Spectral optimization for communication under a peak frequency-domain power constraint

ABSTRACT

System and methods for determining an optimized transmit spectra (spectral distributions of transmission power) for a set of communications channels that experience cross-talk among themselves and for transmitting data on the channels. The transmit spectra are preferably constructed so that largely contiguous frequency bands are allocated to each signaling direction (upstream/downstream) on each communications channel and/or to each channel in the set of channels. In one embodiment, each communications channel is restricted to a maximum time-averaged power. The method preferably includes steps of determining the channel transfer functions of the communications channel, determining interference characteristics of the channels, calculating substantially optimal transmit spectra for the communications channels, and redistributing the frequency bins so that they are contiguously grouped in each transmit spectra. The contiguous groupings allow wider frequency bands for signaling in the channel. In one embodiment, the channel is limited by a “peak-power constraint.”

CONTINUATION INFORMATION

This application is a continuation of:

-   -   U.S. application Ser. No. 09/107,975 titled “Spectral        optimization and joint signaling techniques for communication in        the presence of crosstalk,” by Rohit Gaikwad and Richard        Baraniuk, filed on Jun. 30, 1998, now abandoned and assigned to        the assignee of this application;        which in turn claims the benefit of priority of:    -   U.S. Provisional Application No. 60/068,123 titled “Spectral        optimization and joint signalling techniques for twisted pair        communication,” by Rohit Gaikwad and Richard Baraniuk, filed on        Dec. 19, 1997;    -   U.S. Provisional Application No. 60/083,750 titled “Spectral        optimization and joint signaling techniques for communication in        the presence of crosstalk,” by Rohit Gaikwad and Richard        Baraniuk, filed on Apr. 30, 1998; and    -   U.S. Provisional Application No. 60/087,255 titled “Spectral        optimization and joint signaling techniques for communication in        the presence of crosstalk,” by Rohit Gaikwad and Richard        Baraniuk, filed on May 29, 1998.

MICROFICHE APPENDIX

This application includes a CD appendix. The appendix comprises a sourcecode listing of a preferred embodiment of the invention. A hard copy ofthe appendix is also submitted herewith.

FIELD OF THE INVENTION

The invention relates to electronic communication and, moreparticularly, to techniques for communicating on communications channelssubject to interference such as cross talk and noise.

OUTLINE

Description of the Related Art

1 Communications Background

-   -   1.1 Twisted pairs    -   1.2 Overview of services    -   1.3 Crosstalk interference        -   1.3.1 NEXT and FEXT        -   1.3.2 Notation for self-NEXT and self-FEXT    -   1.4 Capacity and performance margin

2 Problem Statement

-   -   2.1 General statement    -   2.2 Particular statement for DSLs        -   2.2.1 HDSL2 service        -   2.2.2 “GDSL” service        -   2.2.3 “VDSL2” service

3 Previous Work

-   -   3.1 Static PSD Masks and transmit spectra    -   3.2 Joint signaling techniques    -   3.3 Multitone modulation    -   3.4 Summary of previous work        Summary of the Invention        Brief Description of the Drawings        Detailed Description of the Preferred Embodiments

4 New, Optimized Signaling Techniques

-   -   4.1 Assumptions, Notation, and Background    -   4.2 Interference models and simulation conditions    -   4.3 Signaling schemes    -   4.4 Optimization: Interference from other services (DSIN-NEXT        and DSIN-FEXT)—Solution: EQPSD signaling        -   4.4.1 Problem statement        -   4.4.2 Additional assumption        -   4.4.3 Solution        -   4.4.4 Examples    -   4.5 Optimization: Interference from other services (DSIN-NEXT        and DSIN-FEXT) plus self-interference (self-NEXT and low        self-FEXT)—Solution: EQPSD and FDS signaling        -   4.5.1 Self-NEXT and self-FEXT rejection using orthogonal            signaling        -   4.5.2 Problem statement        -   4.5.3 Additional assumptions        -   4.5.4 Signaling scheme        -   4.5.5 Solution: One frequency bin        -   4.5.6 Solution: All frequency bins        -   4.5.7 Algorithm for optimizing the overall transmit spectrum        -   4.5.8 Fast, suboptimal solution for the EQPSD to FDS            switch-over bin        -   4.5.9 Flow of the scheme        -   4.5.10 Grouping of bins and wider subchannels        -   4.5.11 Examples and results        -   4.5.12 Spectral compatibility    -   4.6 Optimization: Interference from other services (DSIN-NEXT        and DSIN-FEXT) plus self-interference (self-NEXT and high        self-FEXT)—Solution: EQPSD, FDS and multi-line FDS signaling        -   4.6.1 Self-FEXT and self-NEXT rejection using multi-line FDS        -   4.6.2 Problem statement        -   4.6.3 Additional assumptions        -   4.6.4 Signaling scheme        -   4.6.5 Solution using EQPSD and FDS signaling: All frequency            bins        -   4.6.6 Switch to multi-line FDS: One frequency bin        -   4.6.7 Switch to multi-line FDS: All frequency bins        -   4.6.8 Special case: Performance of 2 lines        -   4.6.9 Flow of the scheme        -   4.6.10 Examples and results    -   4.7 Joint signaling for lines differing in channel, noise, and        interference characteristics        -   4.7.1 Solution for 2 lines: EQPSD and FDS signaling        -   4.7.2 Solution for M lines: EQPSD and FDS signaling        -   4.7.3 Solution for 2 lines: EQPSD and multi-line FDS            signaling    -   4.8 Optimizing under a PSD mask constraint: No self-interference        -   4.8.1 Problem statement        -   4.8.2 Solution        -   4.8.3 Examples    -   4.9 Optimizing under a PSD mask constraint: With        self-interference        -   4.9.1 Problem statement        -   4.9.2 Solution        -   4.9.3 Algorithm for constrained optimization of the transmit            spectra        -   4.9.4 Examples and results    -   4.10 Bridged taps        -   4.10.1 Optimal transmit spectra        -   4.10.2 Suboptimal transmit spectra        -   4.10.3 Examples and discussion    -   4.11 Optimization: Asymmetrical data-rate channels    -   4.12 Extensions        -   4.12.1 More general signaling techniques        -   4.12.2 More general interferer models        -   4.12.3 Channel variations        -   4.12.4 Broadband modulation schemes        -   4.12.5 Linear power constraints in frequency        -   4.12.6 CDS signaling under a peak power constraint in            frequency        -   4.12.7 Multi-user detector at central office

5 Summary of Contributions

References

Glossary

Notation

DESCRIPTION OF THE RELATED ART 1 Communications Background

1.1 Twisted Pair Telephone Lines

Telephone service is provided to most businesses and homes via a pair ofcopper wires (a “twisted pair”). A telephone cable contains many twistedpairs: 25 twisted pairs are grouped in close proximity into 2 “bindergroups,” and several binder groups are packed together to form a cable.The two terminations of a telephone cable are at the user (subscriber)end and at the telephone company (central office, CO) end. We will usethe terms “twisted pair,” “line,” and “subscriber loop” interchangeablyherein as one example of a communications channel.

Voice telephony uses only the first 4 kHz of bandwidth available on thelines. However, one can modulate data to over 1 MHz with significant bitrates. Only recently have schemes been developed to exploit theadditional bandwidth of the telephone channel. A plot of the frequencyresponse of a typical telephone channel is given in FIG. 1.

1.2 Overview of Services

In the past few years, a number of services have begun to crowd thebandwidth of the telephone channel. Some of the important services are:

-   -   POTS—“Plain Old Telephone Service.” This is the basic telephone        service carrying voice traffic in the 0-4 kHz bandwidth.        Conventional analog modems also use the same bandwidth.    -   ISDN—Integrated Services Digital Network. This service allows        end-to-end digital connectivity at bit rates of up to 128 kbps        (kilo-bits-per-second).    -   T1—Transmission 1. This is a physical transmission standard for        twisted pairs that uses 24 multiplexed channels (each at 64        kbps) to give a total bit rate of 1.544 Mbps        (Mega-bits-per-second). It uses costly repeaters.    -   HDSL—High bit-rate Digital Subscriber Line. This is a        full-duplex (two-way) T1-like (1.544 Mbps) signal transmission        service using only two twisted pairs and no repeaters.    -   ADSL—Asymmetric Digital Subscriber Line. Over one twisted pair,        this service provides a high-speed (on the order of 6 Mbps)        downstream (from central office (CO) to subscriber) channel to        each user and a low-speed (on the order of 640 kbps) upstream        (from subscriber to the central office) channel. This service        preserves the POTS service over a single twisted pair.    -   VDSL—Very high bit-rate DSL. This yet-to-be-standardized service        will provide a very high speed (on the order of 25 Mbps)        downstream channel to subscribers and a lower speed upstream        channel to the central office over a single twisted pair less        than 3 to 6 kft long. Further, it will preserve the POTS        service.    -   HDSL2—High bit-rate Digital Subscriber Line 2. This        soon-to-be-standardized service will provide full-duplex 1.544        Mbps signal transmission service in both directions (full        duplex) over a single twisted pair (<18 kft long) without        repeaters.    -   “GDSL”—General Digital Subscriber Line. This hypothetical        service would (for illustration purposes) carry 25 Mbps        full-duplex data rate over a single twisted pair (see Sections        2.2.2 and 4.6.10).    -   “VDSL2”—Very high bit-rate DSL Line 2. This hypothetical service        would (for illustration purposes) carry 12.4 Mbps full-duplex        data rate over a single twisted pair less than 3 to 6 kft long        (see Sections 2.2.3 and 4.6.10).

Currently, all the above mentioned services have an ANSI standard exceptfor VDSL, HDSL2, “GDSL” and “VDSL2”. The various DSL standards (such asgeneric DSLs, ADSL, VDSL, HDSL2, and others) are collectively referredto as xDSL.

1.3 Crosstalk Interference

1.3.1 NEXT and FEXT

Due to the close proximity of the lines within a binder, there isconsiderable amount of crosstalk interference between differentneighboring telephone lines. Physically, there are two types ofinterference (see in FIG. 2):

-   -   Near-end crosstalk (NEXT): Interference between neighboring        lines that arises when signals are transmitted in opposite        directions. If the neighboring lines carry the same type of        service then the interference is called self-NEXT; otherwise, we        will refer to it as different-service NEXT.    -   Far-end crosstalk (FEXT): Interference between neighboring lines        that arises when signals are transmitted in the same direction.        If the neighboring lines carry the same type of service then the        interference is called self-FEXT; otherwise, we will refer to it        as different-service FEXT.

FIG. 3 shows that crosstalk interference can be modeled as additiveinterference. Since neighboring lines may carry either the same or adifferent flavor of service, there are three categories of interference(see FIG. 3):

-   -   1. Self-interference (self-NEXT and self-FEXT) between lines        carrying the same service.    -   2. Interference into a channel carrying service A from other        lines carrying services other than A (DSIN-NEXT and DSIN-FEXT).    -   3. Interference from a channel carrying service A into other        lines carrying services other than A (DSOUT-NEXT and        DSOUT-FEXT).

Channel noise will be modeled as additive Gaussian noise (AGN).

1.3.2 Notation for Self-NEXT and Self-FEXT

Here is some notation to keep things clear. Number the M twisted pairs(lines) in the cable with index iε{1, . . . , M}, and denote thedirection of transmission with index oε{u, d}, with u=upstream (to thecentral office) and d=downstream (from the central office). All thetwisted pairs in the cable bundle are assumed to carry the same service.Let ō be the complement direction of o: ū=d, {overscore (d)}=u. Denotethe transmitters and receivers on line i as:

-   -   T_(i) ^(o): transmitter (Tx) on twisted pair i in direction o.    -   R_(i) ^(o): receiver (Rx) on twisted pair i in direction o.

Ideally, T_(i) ^(o) intends to transmit information only to R_(i) ^(o).In a real system, however, T_(i) ^(o)'s signal leaks into the receiversR_(j) ^(ō) and R_(j) ^(o). Using our notation, this self-interferencecorresponds to:

-   -   Self-NEXT: Crosstalk from T_(i) ^(o) into R_(j) ^(ō) for all        j≠i, oε{u, d}, and    -   Self-FEXT: Crosstalk from T_(i) ^(o) into R_(j) ^(o) for all        j≠i, oε{u, d}.

In a full-duplex xDSL service, each twisted pair i supports transmissionand reception in both directions (using echo cancelers), so each line ihas a full set of transmitters and receivers: {T_(i) ^(u), R_(i) ^(u),T_(i) ^(d), R_(i) ^(d)}. With perfect echo cancellation, there is nocrosstalk from T_(i) ^(o) into R_(i) ^(ō). We will assume this for thebalance of this document, although this crosstalk could be dealt with ina fashion similar to self-NEXT and self-FEXT.

1.4 Capacity and Performance Margin

The Channel capacity C is defined as the maximum number of bits persecond that can be transmitted over a channel with an arbitrarily smallbit error probability. The achievable rate R_(A) for a channel is anytransmission rate below or equal to capacity, i.e., R_(A)≦C. Anotherchannel performance metric is performance margin (or margin). It isdefined (in dB) as${margin} = {10{\log_{10}\left( \frac{{SNR}_{rec}}{{SNR}_{m\quad i\quad n}} \right)}}$where SNR_(rec) is the received signal-to-noise ratio (SNR) andSNR_(min) is the minimum received SNR required to achieve a fixed biterror probability (BER) at a given transmission rate. The performancemargin of a channel for a fixed bit error probability measures themaximum degradation (from noise and interference) in achievable bit ratethat a channel can sustain before being unable to transmit at that bitrate for a fixed BER (see [12]). The higher the performance margin of achannel at a given transmission rate and fixed BER, the more robust itis to noise and interference, i.e., the better is its performance.

2 Problem Statement

2.1 General Statement

Given an arbitrary communications channel with:

-   -   1. Self-interference (self-NEXT and self-FEXT) between users of        service A,    -   2. Interference from users of different services with users of        service A (DSIN-NEXT and DSIN-FEXT),    -   3. Interference from users of service A into users of different        services (DSOUT-NEXT and DSOUT-FEXT), and    -   4. Other interference (including noise),        maximize the capacity of each user of service A without        significant performance (capacity or margin) degradation of the        other services.

Here services could refer to different possible signaling schemes. Usersrefer to the generic Tx-Rx pairs.

2.2 Particular Statement for DSLs

2.2.1 HDSL2 Service

As a special case of the general problem, we will look into a particularproblem of subscriber loops. In particular, we can phrase our statementin the language of HDSL2 [2]. Here, the communications channel is thecollection of twisted pairs in the telephone cable, interference iscaused by:

-   -   1. Self-NEXT and self-FEXT between neighboring HDSL2 lines        (self-NEXT dominates over self-FEXT [8]),    -   2. DSIN-NEXT and DSIN-FEXT from T1, ISDN, HDSL and ADSL,    -   3. Interference from HDSL2 into other services, such as T1,        ISDN, HDSL and ADSL, and    -   4. Channel noise, which we will model as AGN.

We wish to maximize the capacity of the HDSL2 service in presence ofother HDSL2, T1, ISDN, HDSL, ADSL, VDSL lines and even services not yetimagined while maintaining spectral compatibility with them. We willconsider HDSL2 service in Sections 4.4 to 4.7.

The HDSL2 service is intended to fill a key need for fast (1.544 Mbps)yet affordable full duplex service over a single twisted pair. Effortsto define the standard are being mounted by several companies and theT1E1 standards committee. The two key issues facing HDSL2 standardscommittee are:

-   -   Spectral optimization. All previously proposed schemes for HDSL2        achieve the required data rates with satisfactory margins only        in complete isolation.    -    However, due to the proximity of the lines in a cable, there is        considerable DSIN-NEXT, DSIN-FEXT, self-NEXT and self-FEXT        interference from T1, ISDN, HDSL, ADSL and HDSL2 into HDSL2—this        interference reduces the capacity of the HDSL2 service.    -    Simultaneously, there is considerable DSOUT-NEXT and DSOUT-FEXT        interference from HDSL2 into T1, ISDN, HDSL and ADSL. This        problem is known as spectral compatibility. The scheme        ultimately adopted for HDSL2 must not interfere overly with        other DSL services like T1, ISDN, HDSL, and ADSL.    -    Modulation scheme. No prior system has been developed that        systematically optimizes the HDSL2 spectrum and reduces        interference effects both from and into HDSL2. Further, a        modulation scheme for HDSL2 has not been decided upon at this        time.

2.2.2 “GDSL” Service

Consider the hypothetical DSL service “GDSL” described above. The “GDSL”service will enable very high bit-rate full-duplex, symmetric trafficover a single twisted pair. We assume that the lines carrying GDSLservice have good shielding against self-NEXT. In this case,interference is caused by:

-   -   1. Self-NEXT and self-FEXT between neighboring “GDSL” lines        (self-FEXT dominates over self-NEXT),    -   2. DSIN-NEXT and DSIN-FEXT from T1, ISDN, HDSL, HDSL2 and ADSL,    -   3. Interference from “GDSL”, into other services, such as T1,        ISDN, HDSL, HDSL2 and ADSL, and    -   4. Channel noise, which we will model as AGN.

We wish to maximize the capacity of the “GDSL” service in presence ofother “GDSL”, T1, ISDN, HDSL, ADSL, HDSL2 lines and even services notyet imagined while maintaining spectral compatibility with them. Thespectral optimization issue is similar to the one discussed for HDSL2case, and we need to find an optimal transmit spectrum for “GDSL”.Further, a good modulation scheme needs to be selected.

2.2.3 “VDSL2” Service

Consider the hypothetical DSL service “VDSL2” described above. Opticalfiber lines having very high channel capacity and virtually no crosstalkwill be installed in the future up to the curb of each neighborhood(FTTC). The final few thousand feet up to the customer premises could becovered by twisted pairs. In such a scenario, high bit-rateasymmetric-traffic services (like VDSL) and symmetric-traffic services(like “VDSL2”) over short length twisted pairs would become important.For illustration of such a potential future service we propose ahypothetical “VDSL2” service that would carry very high bit-ratesymmetric traffic over short distance loops on a single twisted pair. Inthe “VDSL2” case, the interference will be caused by:

-   -   1. Self-NEXT and self-FEXT between neighboring “VDSL2” lines        (both self-NEXT and self-FEXT are dominant),    -   2. DSIN-NEXT and DSIN-FEXT from T1, ISDN, HDSL, HDSL2, VDSL and        ADSL,    -   3. Interference from “VDSL2” into other services, such as T1,        ISDN, HDSL, HDSL2, VDSL and ADSL, and    -   4. Channel noise, which we will model as AGN.

Again, we wish to maximize the capacity of “VDSL2” in presence of allthe other interferers. To achieve this we need to find optimal transmitspectra and a good modulation scheme.

3 Previous Work

Here we discuss prior work pertaining to HDSL2 service.

3.1 Static PSD Masks and Transmit Spectra

The distribution of signal energy over frequency is known as the powerspectral density (PSD). A PSD mask defines the maximum allowable PSD fora service in presence of any interference combination. The transmitspectrum for a service refers to the PSD of the transmitted signal.Attempts have been made by several groups to come up with PSD masks forHDSL2 that are robust to both self-interference and interference fromother lines. One way of evaluating channel performance is by fixing thebit rate and measuring the performance margins [12]: The higher theperformance margin for a given disturber combination, the more robustthe HDSL2 service to that interference. The term crosstalk here impliesself-interference plus interference from other lines.

To the best of our knowledge, no one has optimized the PSD of HDSL2lines in presence of crosstalk and AGN. The significant contributions inthis area, MONET-PAM and OPTIS, [1, 2, 4, 5] suggest a staticasymmetrical (in input power) PSD mask in order to attempt to suppressdifferent interferers. The PSD masks suggested in [1, 2, 4, 5] have adifferent mask for each direction of transmission. Furthermore, thetechniques in [1, 4] use different upstream and downstream averagepowers for signal transmission. However, the mask is static, implying itdoes not change for differing combinations of interferers.

Optis [5] is currently the performance standard for HDSL2 service.

When a constraining PSD mask is imposed, the transmit spectrum liesbelow the constraining mask. Specifying a constraining PSD mask onlylimits the peak transmit spectrum. We do PSDs (transmit spectra) and notmasks in this document unless stated otherwise. In Section 4.11 weindicate ideas to get PSD masks.

3.2 Joint Signaling Techniques

Self-NEXT is the dominant self-interference component insymmetric-data-rate, full-duplex, long-length line xDSL service (e.g.,HDSL2). One simple way of completely suppressing self-NEXT is to useorthogonal signaling (for example, time division signaling (TDS),frequency division signaling (FDS), or code division signaling (CDS)).In TDS, we assign different services to different time slots. In FDS, weseparate in frequency the services that could interfere with each other.In CDS, a unique code or signature is used in each direction of service.Further, in CDS each service occupies the entire available bandwidth forall of the time. CDS is similar to code division multiple access (CDMA),but here instead of providing multiple access, CDS separates theupstream and downstream transmit spectra using different codes.

The choice of orthogonal signaling scheme depends on the intent. We willsee that FDS is in a sense optimal under an average power constraint(see Section 4.5.12).

To eliminate self-NEXT using FDS, we would force the upstreamtransmitters {T_(i) ^(u), i=1, . . . , M}, and the downstreamtransmitters {T_(i) ^(d), i=1, . . . , M} to use disjoint frequencybands. Thus, in FDS signaling, the upstream and downstream transmissionsare orthogonal and hence can be easily separated by the correspondingreceivers. Since in a typical system FDS cuts the bandwidth available toeach transmitter to ½ the overall channel bandwidth, we have anengineering tradeoff: FDS eliminates self-NEXT and therefore increasessystem capacity; however, FDS also reduces the bandwidth available toeach transmitter/receiver pair and therefore decreases system capacity.When self-NEXT is not severe enough to warrant FDS, both upstream anddownstream transmitters occupy the entire bandwidth. In this case, theupstream and downstream directions have the same transmit spectrum; werefer to this as equal PSD (EQPSD) signaling.

On a typical telephone channel, the severity of self-NEXT varies withfrequency. Therefore, to maximize capacity, we may wish to switchbetween FDS and EQPSD depending on the severity of self-NEXT. Such ajoint signaling strategy for optimizing the performance in the presenceof self-NEXT and white AGN was introduced in [3].

The scheme in [3] is optimized, but only for an over simplified scenario(and therefore not useful in practice). In particular, [3] does notaddress self-FEXT and interference from other lines as considered inthis work. Further, [3] does not address spectral compatibility issue.

All other schemes for joint signaling employ ad-hoc techniques forinterference suppression [1, 2, 4, 5].

3.3 Multitone Modulation

Multicarrier or discrete multitone (DMT) modulation [6] can be readilyused to implement a communication system using a wide variety of PSDs.Multitone modulation modulates data over multiple carriers and adjuststhe bit rate carried over each carrier according to the signal to noiseratio (SNR) for that carrier so as to achieve equal bit errorprobability (BER) for each carrier (see in FIG. 4).

Orthogonal FDS signaling is easily implemented using the DMT: we simplyassign transmitter/receiver pairs to distinct sets of carriers. Note,however, that multitone modulation is definitely not the only modulationscheme that can be used to implement (optimal) transmit spectra. We canjust as well use other techniques, such as CAP, QAM, multi-level PAM,etc.

3.4 Summary of Previous Work

The current state of the art of DSL technology in general and HDSL2 inparticular can be described as follows:

-   -   Ad-hoc schemes (sometimes referred to as “optimized”) have been        developed that attempt to deal with self-interference and        DSIN-NEXT and DSIN-FEXT as well as spectral compatibility of the        designed service with other services. However, these schemes by        no means optimize the capacity of the services considered.    -   An optimal signaling scheme has been developed in [3] for the        case of self-NEXT and white additive Gaussian noise only. The        development of [3] does not address crosstalk from other        sources, such as DSIN-NEXT and DSIN-FEXT, or self-FEXT, or other        types of additive Gaussian noise. The development of [3] also        does not address spectral compatibility of the designed service        with respect to other services.

SUMMARY OF THE INVENTION

Described herein are methods for constructing optimized transmit spectra(spectral distributions of transmission power) for signaling on a set ofdigital communications channels that experience cross-talk amongthemselves and/or between upstream and downstream transmission. Thetransmit spectra are preferably constructed so that largely contiguousfrequency bands are allocated to each signaling direction(upstream/downstream) on each channel and/or to each channel in the setof channels.

In one embodiment, each communications channel is restricted to amaximum time-averaged power. The method preferably includes steps ofdetermining the channel transfer functions of the communicationschannels, determining interference characteristics of the channels, andthen calculating substantially optimal transmit spectra for thecommunications channels. Two transmit spectra are preferably constructedfor each communications channel: one for signaling in each direction. Ingeneral, the optimized transmit spectra include portions of thefrequency spectrum where upstream transmission is orthogonally separatedfrom downstream transmission (frequency-division signaling or FDS). Thatis, these portions are divided into a number of “bins,” and eitherupstream transmission or downstream transmission—but not both—occurs ina given frequency bin. Similarly, the transmit spectra may includeregions where the individual channels are orthogonally separated fromeach other (multi-line FDS). The next step in the method isre-distributing the bins so that they are contiguously grouped in eachtransmit spectrum. The contiguous groupings allow wider frequency bandsfor signaling in the channel.

The re-distributions are preferably done in such a manner that thechannel capacities are matched for the two signaling directions and/orfor each communications channel. They may also or alternatively beperformed so that the performance margins or the maximum time-averagedsignaling powers are matched.

Also described herein are a method and a system for communicating dataon a communications channel limited by a “peak-power constraint.” Thepeak power constraint specifies maximum power limits in one or morespectral regions for the channel. The method determines a transmitspectrum in response to the peak power constraint and other factors,such as the channel's transfer function and interference characteristicsof the channel. Data are then preferably transmitted in a signal with aspectral distribution given by the transmit spectrum. The signal may bea discrete multitone signal, for example, or another signal with aspectral distribution or PSD (power spectral density) given by thetransmit spectrum.

In one embodiment the method includes the steps of determining thechannel transfer function for the communications channel, determiningthe interference characteristics of the channel, and determining thetransmit spectrum in response to the channel transfer function and theinterference characteristics.

In another embodiment, the method determines transmit spectra for a setof communications channels. The transmit spectra are preferablyoptimized for capacity given the effects of cross-talk among thechannels and other interferences. In this embodiment, the methodcomprises the steps of determining channel transfer functions of thecommunications channels, determining their interference characteristics,and then determining the transmit spectra in response to the channeltransfer functions and the interference characteristics.

The transmit spectrum is preferably determined by a minimization ormaximization procedure, such as a water-filling technique, to maximizethe capacity of the communications channels. With the consideration ofthe peak power constraint, the procedure may be a peak-constrainedwater-filling technique. For each communications channel, the transmitspectra preferably comprise an upstream transmit spectrum and adownstream transmit spectrum, which are used for transmission inopposite directions on the channel. To minimize the effects ofinterference between signals traveling in the same direction (near-endcross-talk, or NEXT), the transmit spectra may be determined so thatupstream communications are orthogonally separated from downstreamcommunications in one or more spectral regions. To minimize the effectsof interference between signals traveling in opposite directions(far-end cross-talk, or FEXT), the transmit spectra may be determined sothat each channel is orthogonally separated, in one or more spectralregions, from other channels in the set of communications channels. Inone embodiment, transmission on the communications channels is limitedto a predetermined time-averaged power. The transmit spectra are thendetermined in response to this average power constraint.

BRIEF DESCRIPTION OF THE DRAWINGS

Other objects and advantages of the invention will become apparent uponreading the following detailed description and upon reference to theaccompanying drawings in which:

FIG. 1 is an example of the frequency-response for a twisted pairtelephone channel;

FIG. 2 shows NEXT and FEXT between neighboring lines in a telephonecable, with “Tx” and “Rx” indicating transmitters and receivers,respectively;

FIG. 3 shows how NEXT (DSIN-NEXT and self-NEXT) and FEXT (DSIN-FEXT andself-FEXT) are modeled as additive interference sources, with DSOUT-NEXTand DSOUT-FEXT representing the interference leaking out into otherneighboring services;

FIG. 4 illustrates how multicarrier, or discrete multitone (DMT)modulation multiplexes the data onto multiple orthogonal carrier waves;

FIG. 5 and FIG. 5A are representative views of a subscriber-linecommunications system and a well-logging system that use the presentinvention;

FIG. 6 is a representative view of a home system using the presentinvention for DSL communications;

FIG. 7 is a block diagram of one embodiment of the computer from FIG. 6;

FIG. 8 is a block diagram of one embodiment of the DSL card from FIG. 7;

FIG. 9 is a flowchart for determining transmission characteristics for acommunications system in one embodiment of the invention;

FIG. 10 is a flowchart for determining a transmit spectrum withpreliminary analyses of self interference and FEXT levels;

FIG. 11 is a flowchart for determining a transmit spectrum withpreliminary analyses of self interference levels;

FIG. 12 is a flowchart for determining a transmit spectrum;

FIG. 13 is a flowchart for method for transmitting data on acommunications channel;

FIG. 14 is a flowchart for initiating a data transfer on thecommunications channel;

FIG. 15 is a flowchart for determining transmission characteristics fora communications system in one embodiment of the invention;

FIG. 16 is a frequency-response graph showing the channel sub-divisioninto K narrow bins (subchannels), each of width W (Hz);

FIG. 17 shows the magnitude squared transfer function of the channel(CSA loop 6), with 39 self-NEXT interferers, and 39 self-FEXTinterferers (see (1)-(3));

FIG. 18 shows transmit spectra for EQPSD, FDS and multi-line FDSsignaling schemes in a single frequency bin k for the case where thenumber of lines is 3 (this also works for any number of lines);

FIG. 19 is a model for combined additive interference from otherservices (DSIN-NEXT and DSIN-FEXT) plus channel noise (AGN);

FIG. 20 is a flowchart of a method for determining an optimal transmitspectrum using only EQPSD signaling;

FIG. 21 is a graph of an optimal transmit spectrum of HDSL2 (on CSA loop6) with 49 HDSL DSIN-NEXT interferers and AGN of −140 dBm/Hz;

FIG. 22 is a graph of an optimal transmit spectrum of HDSL2 (on CSA loop6) with 25 T1 DSIN-NEXT interferers and AGN of −140 dBm/Hz;

FIG. 23 shows upstream and downstream transmit spectra in a singlefrequency bin (α=0.5→EQPSD signaling and α=1→FDS signaling);

FIG. 24 is a graph demonstrating that RA is monotonic in the intervalαε(0.5, 1];

FIG. 25 shows EQPSD and FDS signaling in a single frequency bin;

FIG. 26 shows upstream and downstream transmit spectra with regionsemploying EQPSD signaling (in bins [1, M_(E2F)]) and FDS signaling (inbins [M_(E2F)+1, K]);

FIG. 27 is a flowchart of the optimal and suboptimal schemes todetermine the transmit spectrum using EQPSD and FDS signaling (andEQPSD/FDS transmit spectrum);

FIG. 28 shows joint EQPSD/FDS signaling for a channel with “discrete”and “contiguous” transmit spectra for upstream (top graphs) anddownstream (bottom graphs) signaling;

FIG. 29 is a graph of an optimal upstream transmit spectrum for CSA Loop6 using HDSL2 with 39 self-NEXT and 39 self-FEXT interferers, with EQPSDsignaling taking place to the left of bin 9 (indicated by solid line)and FDS signaling taking place to the right (indicated by dashed line);

FIG. 30 shows graphs of optimal “contiguous” upstream and downstreamtransmit spectra for CSA Loop 6 using HDSL2 with 39 self-NEXT and 39self-FEXT interferers (EQPSD signaling taking place to the left of bin9);

FIG. 31 shows graphs of another set of optimal “contiguous” upstream anddownstream transmit spectra for CSA Loop 6 using HDSL2 with 39 self-NEXTand 39 self-FEXT interferers; with the property that these spectra yieldequal performance margins (equal capacities) and equal average powers inboth directions of transmission (EQPSD signaling taking place to theleft of bin 9);

FIG. 32 shows transmit spectra of signaling line (S), interfering line(Y and Z), and lumped channel noise (N) for two cases: the FDS scheme(Case 2) for interfering line yields higher capacity for signaling line(S) than other schemes like CDS (Case 1);

FIG. 33 shows EQPSD and multi-line FDS signaling in a single frequencybin k for the M=3 line case;

FIG. 34 shows FDS and multi-line FDS signaling in a single frequency bink for the M=3 line case;

FIG. 35 is an example of an upstream transmit spectrum of line 1 (S₁^(u)(f)) employing EQPSD, FDS and multi-line FDS signaling schemes forthe M=3 line case, in which bins [1, M_(E2MFDS)] employ EQPSD, bins[M_(E2MFDS)+1, M_(MFDS2FDS)] employ multi-line FDS, bins[M_(MFDS2FDS)+1, M_(FDS2MFDS)] employ FDS, and bins [M_(FDS2MFDS)+1, K]employ multi-line FDS; The downstream spectrum of line 1 (S₁ ^(d)(f)) issimilar to S₁ ^(u)(f) except for-putting power in the complementaryhalves of FDS bins; The upstream spectra of lines 2 and 3 are similar toS₁ ^(u)(f) except for putting power in complementary thirds ofmulti-line FDS bins; The downstream spectra for lines 2 and 3 aresimilar to S₁ ^(u)(f) except for putting power in the complementaryhalves of the FDS bins and in the complementary thirds of multi-line FDSbins;

FIG. 36 illustrates practical observation 1, a case of an EQPSD/FDS/MFDStransmit spectrum in which there is no FDS spectral region; bins [1,M_(E2MFDS)] employ EQPSD, and bins [M_(E2MFDS)+1, K) employ multi-lineFDS;

FIG. 37 illustrates practical observation 2, a case of an EQPSD/FDS/MFDStransmit spectrum in which there is no multi-line FDS spectral portionwithin the EQPSD region; bins [1, M_(MFDS2FDS)] employ EQPSD, bins[M_(MFDS2FDS)+1, M_(FDS2MFDS)] employ FDS, and bins [M_(FDS2MFDS)+1, K)employ multi-line FDS;

FIG. 38 shows upstream and downstream transmit spectra in a singlefrequency bin (α=0.5→EQPSD signaling and α=1→multi-line FDS signaling);

FIG. 39 shows EQPSD and multi-line FDS signaling in a single frequencybin;

FIG. 40 is a flowchart of a scheme for determining an optimal transmitspectrum using EQPSD, FDS, and multi-line FDS signaling (anEQPSD/FDS/MFDS transmit spectrum);

FIG. 41 shows, for the case where the lines have different linecharacteristics, upstream and downstream transmit spectra in a singlefrequency bin (α=0.5→EQPSD signaling and α=1→multi-line FDS signaling);

FIG. 42 shows, for the case where the lines have different linecharacteristics, upstream and downstream transmit spectra in a singlefrequency bin (α=0.5→EQPSD signaling and α=1→multi-line FDS signaling);

FIG. 43 is a graph of an optimal downstream transmit spectrum for HDSL2(on CSA loop 6) under an OPTIS downstream constraining PSD mask with 49HDSL DSIN-NEXT interferers and AGN of −140 dBm/Hz (the ‘o—o’ line showsthe peak-constrained optimal transmit spectrum and the ‘—’ line showsthe constraining OPTIS PSD mask);

FIG. 44 is a graph of an optimal upstream transmit spectrum for HDSL2(on CSA loop 6) under an OPTIS upstream constraining PSD mask with 25 T1DSIN-NEXT interferers and AGN of −140 dBm/Hz (the ‘o—o’ line shows thepeak-constrained optimal transmit spectrum and the ‘—’ line shows theconstraining OPTIS PSD mask);

FIG. 45 shows graphs of optimal upstream and downstream transmit spectrafor HDSL2 (on CSA loop 6) under the OPTIS upstream and downstreamconstraining PSD masks with 39 HDSL2 self-NEXT and self-FEXT interferersand AGN of −140 dBm/Hz (the ‘o—o’ lines show the peak-constrainedoptimal transmit spectra and the ‘—’ lines show the constraining OPTISPSD masks);

FIG. 46 shows graphs of optimal upstream and downstream transmit spectrafor HDSL2 (on CSA loop 6) under the OPTIS upstream and downstreamconstraining PSD masks with 24 HDSL2 self-NEXT and self-FEXTinterferers, 24 T1 interferers, and AGN of −140 dBm/Hz (the ‘o—o’ linesshow the peak-constrained optimal transmit spectra and the ‘—’ linesshow the constraining OPTIS PSD masks);

FIG. 47 shows graphs of optimal “contiguous” upstream and downstreamtransmit spectra for HDSL2 (on CSA loop 4, with a non-monotonic channelfunction due to bridged taps) with 39 HDSL2 self-NEXT and self-FEXTinterferers; these transmit spectra yield equal performance margins(equal capacities) and equal average powers in both directions oftransmission (note that there is only one transition region from EQPSDto FDS signaling);

FIG. 48 shows (in the top graph) the channel transfer function,self-NEXT, and self-FEXT transfer functions for a short loop withbridged taps employing “GDSL” service (note that self-NEXT is very lowfor this hypothetical service), and shows (in the bottom graph) thedistributed EQPSD and FDS spectral regions for the upstream anddownstream transmit spectra, with a 0 indicating EQPSD signaling, a 1indicating FDS, and a 0.5 indicating EQPSD or FDS signaling (note thatin this case the non-monotonicity of the channel transfer function leadsto several distributed signaling regions); and

FIG. 49 shows an alternative signaling scheme: in the presence of highdegrees of self-NEXT and self FEXT between group of lines 1 and 2 andlines 3 and 4, we employ multi-line FDS; there is EQPSD signaling withineach group of lines (1 and 2 employ EQPSD as do 3 and 4) that have lowself-interference.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof are shown by way ofexample in the drawings and will herein be described in detail. Itshould be understood, however, that the drawings and detaileddescription thereto are not intended to limit the invention to theparticular form disclosed, but on the contrary, the intention is tocover all modifications, equivalents and alternatives falling within thespirit and scope of the present invention as defined by the appendedclaims.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention comprises an improved system and method forcommunicating information such as voice, images, video, data, or otherinformation on a transmission medium. The present invention providesimproved communications on the transmission medium in the presence ofinterference. More specifically, the present invention operates to modeland then minimize the effects of interference on the transmissionmedium. The interference may take the form of similar services beingtransmitted on neighboring transmission mediums and/or may take the formof uncorrelated interference from different services on neighboringtransmission mediums and or interference from various noise sourceswhich affect the transmission medium.

FIG. 5—Subscriber Line Embodiment

FIG. 5 illustrates a preferred embodiment for use of the presentinvention. FIG. 5 illustrates a first location, e.g., a home 10 that iscoupled through a subscriber line 12 to a second location, e.g. atelephone company central office (CO) 14. It is noted that the first andsecond locations may be any of various types of sites, such as a home,an office, business, an office building, or a CO.

In this embodiment of the invention, the communication system and methodis comprised in a digital subscriber line (DSL) device that operates toperform xDSL communications on subscriber line 12. Thus, this figureshows a configuration that includes subscriber line 12, e.g., atwisted-pair copper line, that is coupled between home 10 and CO 14. Thepresent invention is comprised in each of home 10 and CO 14.

As discussed in the background section, subscriber lines are generallyincluded in a cable that has a plurality of closely positionedtransmission mediums, including other subscriber lines. Due to the closeproximity of the transmission mediums comprised in a subscriber cable, agiven subscriber line is subject to interference from neighboringtransmission mediums, including self-NEXT and self-FEXT interference,and different service interference (DSIN). Some of the transmit spectradiscussed herein are substantially optimized to maximize performancemargins and avoid the effects of this interference, thereby providingimproved communication.

By design, an optimal transmit spectra give increased performancemargins (increased immunity against noise) and spectral compatibilitymargins as compared to one fixed transmit spectrum. The optimal transmitspectra described herein are typically obtained by fixing the averageinput power and choosing the best signaling strategies and optimal powerdistribution to maximize the bit rate or performance margin. Thetransmit spectra may also be used to minimize the required average inputpower where the desired performance margins or bit rates are fixed.

FIG. 5A—Well Logging Embodiment

FIG. 5A illustrates an alternate scenario for use of the communicationsystem and method of the present invention. FIG. 5A illustrates a drillhole and/or well-logging scenario which utilizes the communicationsystem of the present invention. As an example, in FIG. 5A communicationequipment 16 on the surface communicates through a communication medium12A to instrumentation 18 comprised in the borehole underground. Thecommunication system and method operates to reduce the effects ofinterference in the well hole and to provide improved communications.

Although FIG. 5 and FIG. 5A illustrate two embodiments for use of thesystem and method of the present invention, it is noted that the presentinvention may be used in any of various types of systems or scenarioswhich involve communication of data on a transmission medium that issubject to noise or other interference. The present invention isparticularly useful in scenarios where the transmission medium is inclose proximity to various sources of interference that can beascertained, identified, and modeled. In general, the present inventionis applicable to reduce the effects of interference on transmissionmedia that are subject to interference from known or unknown sourceswhere the spectral characteristics of the interference can be modeled.

In the analyses presented herein, we use a generic DSL (xDSL) model. Forconcreteness, we present results optimizing DSL services such as HDSL2,“GDSL”, and “VDSL2” in the face of noise and interference fromneighboring services. The invention is not, however, limited to theseservices, but can be applied to any communications channel that exhibitscrosstalk interference.

Although FIG. 5 illustrates a subscriber line embodiment, it is notedthat the present invention may be used for any of various types oftransmission media, e.g., copper wire, fiber optic, lines, co-axialcable, wave guides, etc. For example, the present invention is wellsuited for use in local and wide-area networks to minimize noiseinterference on networks, e.g., Ethernet, token ring, and wide areanetworks such as frame relay, Switched 56, ATM (asynchronous transfermode), etc. Also, although FIG. 5 illustrates use of the presentinvention between a home 10 and a central office 14 over a subscriberloop or subscriber line 12, it is noted that the present invention mayalso be used for the various trunks comprised in the PSTN. The presentinvention is also useful in the various backbones or lines used for theInternet.

The present invention is also useful for wireless transmissionapplications, e.g., cellular telephones, cordless telephones, short waveradio, etc. as well as the various broadcast media such as the variousdigital satellite services providing television, Internet, data or voiceservices. In short, the present invention is applicable to any ofvarious types of systems which involve the transmission of data over awired or wireless medium. The present invention is also applicable toany of the various types of signaling protocols such as frequencydivision multiple access (FDMA), time division multiple access (TDMA)and code division multiple access (CDMA) as well as hybrids of these,among others.

Therefore, the present invention is applicable to a variety ofcommunications channels in a number of communication scenarios. In thedescription that follows, the present invention is described withrespect to the preferred embodiment, the preferred embodiment being adigital subscriber line application between a first location, e.g., ahome or business 10, and a telephone company central office 14.

FIG. 6—Home DSL System

FIG. 6 illustrates a system 100 comprised in location 10, i.e., in thehome or business 10 which performs digital subscriber line (DSL)communication operations over subscriber line 12. In a preferredembodiment, the DSL circuitry of the present invention is comprised in acomputer system 102 coupled to subscriber line 12 through an xDSL port106. In one embodiment, computer system 102 is also coupled to atelephone system 104. However, it is noted that the DSL system of thepresent invention may be comprised in any of various types of systemsincluding computer systems, Internet appliances, televisions ordedicated boxes. In the preferred embodiment, the DSL system of thepresent invention is comprised in a DSL device on an add-in card to thegeneral purpose computer system 102. In the preferred embodiment, theDSL card includes a port for coupling to a standard telephone jack, or“splitter,” which in turn couples to the subscriber line 12. In thisembodiment, the computer system 102 may be utilized as a virtualtelephone which operates through the DSL device for voice communicationsover the subscriber line 12. In another embodiment, a separate telephonesystem 104 is coupled to a second port of the DSL card, as shown in FIG.6.

FIG. 7—Computer System Block Diagram

Turning now to FIG. 7, a block diagram of one embodiment of computersystem 102 is shown. Other embodiments are possible and contemplated.The depicted system includes a microprocessor or CPU 110 coupled to avariety of system components through a bus bridge 114. In the depictedsystem, a main memory 112 is also coupled to bus bridge 114. Finally, aplurality of PCI devices are coupled to bus bridge 114 through a PCI bus116. In the depicted embodiment, the PCI devices include a video card118 and a add-in card for the DSL device 120.

Bus bridge 114 provides an interface between microprocessor 110, mainmemory 112, and the devices attached to PCI bus 116. When an operationis received from one of the devices connected to bus bridge 114, busbridge 114 identifies the target of the operation (e.g. a particulardevice or, in the case of PCI bus 116, that the target is on PCI bus116). Bus bridge 114 routes the operation to the targeted device. Busbridge 114 generally translates an operation from the protocol used bythe source device or bus to the protocol used by the target device orbus.

Main memory 112 is a memory in which application programs are stored andfrom which microprocessor 110 primarily executes. A suitable main memory112 comprises DRAM (Dynamic Random Access Memory), and preferably aplurality of banks of SDRAM (Synchronous DRAM).

FIG. 8—DSL Device

FIG. 8 is a block diagram illustrating DSL device 120 comprised in thecomputer 102 of FIG. 6. As noted above, although in the preferredembodiment the DSL device is comprised as a computer add-in card, DSLdevice 120 may take any of various types of forms including beingcomprised in a television system, Internet appliance, or dedicateddevice, among others systems. As shown, the DSL device or add-in card120 comprises a first port 160 for coupling to an expansion bus of thecomputer system, preferably a PCI expansion bus port as shown. DSLdevice 120 also includes at least one subscriber line port 170 forcoupling to the digital subscriber line 12. DSL device 120 may includeany of various hardware elements for performing the communicationoperations of the present invention. For example, in one embodiment, theDSL communication device includes one or more programmable processingunits which implement instructions from a memory. For example, the DSLcommunication device may include a programmable digital signal processor(DSP) 152, a general purpose processor, or other processors that executeinstructions from a memory 156 to implement the communication operationsof the present invention. Alternatively, or in addition, the DSLcommunication device 120 includes one or more application specificintegrated circuits (ASICs) 154 and 158 or programmable logic devicessuch as FPGAs etc. that implement a portion or all of the presentinvention. In short, the communication system and method of the presentinvention may be implemented in any of various types of ways includingprogrammable devices such as processing units, CPUs, DSPs,microcontrollers, etc., dedicated hardware such as ASICs, programmablelogic devices such as FPGAs, or combinations of the above.

FIG. 9-FIG. 12—Method for Determining Transmission Characteristics

FIG. 9 is a flowchart for determining a transmit spectrum for use incommunicating data on a communications channel according to oneembodiment of the present invention. This method may be used incommunicating data on the communications channel when the communicationschannel is subject to interference from one or more other communicationschannels. The communications channel of interest and one or more of theother communications channels carry a particular type of service, suchas xDSL, ISDN, T1, or spread-spectrum, for example. The first steps inthis method comprise determining a channel transfer function of thecommunications channel 210 and an amount of self interference 220 intothe communications channel from the other communications channels thatcarry the same type of service. In step 230, the transfer function andthe amount of self interference are examined, and in step 240 a transmitspectrum for the channel is determined based on the examining.

Determining the channel transfer function in step 210 of FIG. 9 may bedone by directly measuring it. For example, a transmitter on one end ofthe communications channel, such as at CO 12 (in FIG. 1) or inwell-logging instrumentation 18 (in FIG. 1A), may be directed to send asignal or a series of signals with predetermined intensities as afunction of frequency, with which the a receiver at the other end of thechannel may measure the attenuation, and perhaps also the phase shift,as a function of frequency. The measurement may be extended to determinenonlinear responses of the channel by repeating the measurement withvarying source strengths. Alternately, the channel characteristics maybe determined in advance of the communication and stored, for example ina database at the CO or in a memory on a DSL card. Determining thechannel transfer function could then entail receiving it from the CO,the local memory storage, or other storage locations. In the case wherethe invention is used in a subscriber line system, receiving the channeltransfer function from the CO is particularly useful since the CO mayrapidly look up pre-stored information on the particular physical linebeing used for the communications channel.

Similarly, the amount of self interference may be determined in step 220of FIG. 9 by receiving it or, if transmitter/receiver pairs areaccessible on the other same-service channels, by measuring it.Determining the amount of self interference in step 220 may comprisedetermining a total self interference power, or a power distribution, ora coupling coefficient from the other same-service channels into thechannel of interest, or a coupling coefficient with frequency dependence(such as a self-interference transfer function) from the othersame-service channels into the channel of interest, or a combination ofthese characteristics, among others. The amount of self interference mayalso include a characterization of the self interference in terms ofself-NEXT and self-FEXT interference. In a preferred embodiment of theinvention, step 220 includes determining a self-NEXT transfer functionand a self-FEXT transfer function from the other same-service channelsinto the channel of interest.

FIG. 10-FIG. 12 show various embodiments 240 a-c of step 240, in whichthe transmit spectrum is determined.

In one embodiment of the invention, as shown in steps 241 and 245 ofFIG. 10, determining the transmit spectrum in step 240 comprisesdetermining an EQPSD transmit spectrum if the amount of selfinterference is substantially low or negligible. An EQPSD transmitspectrum is a transmit spectrum in which EQPSD signaling is used on atleast one portion of the available spectrum of communicationfrequencies.

The method may also include steps 242 and 247, in which an EQPSD/FDStransmit spectrum is found if the amount of self interference issubstantially high or non-negligible.

In general, an EQPSD/FDS transmit spectrum has a number of frequencyregions in which EQPSD signaling is used, and a number of frequencyregions in which FDS signaling is used. The locations of these regionsin the available spectrum of communication frequencies and thetransmission power as a function of frequency are preferably determinedso that the data transmission rate on the channel is substantiallymaximized. An EQPSD/FDS transmit spectrum preferably includes at leastone portion using FDS signaling and one portion using FDS signaling, butin a degenerate case, the maximization may be achieved by using onlyEQPSD or only FDS signaling. An EQPSD/FDS transmit spectrum is thus atransmit spectrum in which the available spectrum of communicationfrequencies includes at least one portion using EQPSD signaling or FDSsignaling.

Still further, the method may also include step 249, in which anEQPSD/FDS/MFDS transmit spectrum is found if the amount of selfinterference is substantially high or non-negligible and if the amountof self-FEXT interference is substantially high. In multi-line FDS(MFDS) signaling, different channels carrying the same or similarservices are orthogonally separated to reduce crosstalk interference.

In general, an EQPSD/FDS/MFDS transmit spectrum has a number offrequency regions in which EQPSD signaling is used, a number offrequency regions in which FDS signaling is used, and a number offrequency regions in which MFDS signaling is used. Again, the locationsof these regions in the available spectrum of communication frequenciesand the transmission power as a function of frequency are preferablydetermined so that the data transmission rate on the channel issubstantially maximized. An EQPSD/FDS/MFDS transmit spectrum preferablyincludes at least one portion using FDS signaling, one portion using FDSsignaling, and one portion using MFDS signaling, but in a degeneratecase, the maximization may be achieved by using only EQPSD or only FDSor only MFDS signaling. An EQPSD/FDS/MFDS transmit spectrum is thus atransmit spectrum in which the available spectrum of communicationfrequencies includes at least one portion using EQPSD signaling or FDSsignaling or MFDS signaling.

In another embodiment of the invention, as shown in FIG. 11, determiningthe transmit spectrum in step 240 comprises determining an EQPSDtransmit spectrum (in step 245) if the amount of self interference issubstantially low or negligible (according to step 241), and determiningan EQPSD/FDS/MFDS transmit spectrum (in step 249) if the amount of selfinterference is substantially high or non-negligible.

In yet another embodiment of the invention, as shown in FIG. 12,determining the transmit spectrum in step 240 comprises determining anEQPSD/FDS/MFDS transmit spectrum (in step 249). For the degenerate casesin which only one or two of the EQPSD, FDS, and MFDS signalingtechniques are needed for maximizing the data transmission rate, theEQPSD/FDS/MFDS transmit spectrum will reduce to the appropriatesignaling techniques.

In a preferred embodiment of the invention, the method further comprisesa step of determining an amount of uncorrelated interference, such asadditive Gaussian noise (AGN), into the communications channel. If oneor more of the other communications channels carry a different type ofservice than the service on the communications channel, then theuncorrelated interference may include different-service interference(DSIN) from the other communications channels carrying the differentservice. Thus, the uncorrelated interference includes a total noiseinterference that preferably comprises AGN, DSIN, and other noise andinterference whose spectral characteristics are not controlled by theuser. Determining the transmit spectrum in step 240 is then performed inresponse to the amount of uncorrelated interference.

FIG. 13-FIG. 14—Method for Transmitting Data

FIG. 13 is a flowchart of a method for transmitting data according toone embodiment of the present invention. This method may be used incommunicating data on a communications channel when the communicationschannel is subject to interference from one or more other communicationschannels. The other communications channels may be located proximate tothe communications channel, for example, in the case of multiplesubscriber lines in a binder group of a telephone cable, or in the caseof multiple radio transmission systems with closely located transmittersor overlapping coverage regions. The communications channel of interestcarries a particular type of service, such as xDSL, ISDN, T1, orspread-spectrum, for example. The method comprises the steps ofdetermining a channel transfer function of the communications channel instep 310, initiating a data transfer on the communications channel instep 320, and transferring the data on the communications channel usingthe transmit spectrum in step 330. As shown in FIG. 14, the step 320 ofinitiating the transfer comprises determining interferencecharacteristics of the interfering communications channels in step 322,and determining a transmit spectrum in response to the channel transferfunction and the interference characteristics in step 324.

As discussed earlier, step 310 of determining the channel transferfunction may comprise measuring the channel transfer function, receivingthe channel transfer function, or determining the channel transferfunction through other means. The channel transfer function may bedetermined at power-up of a transmission system, or at regular intervalsin time, or in response to temperature changes, or at other appropriatetimes.

The transmit spectrum is preferably determined in step 324 tosubstantially maximize the data transmission rate for the communicationschannel, so that the maximum information may be communicated per unittime on the communications channel in light of the various sources ofnoise and interference. The transmit spectrum is also preferablydetermined in such a manner that the communications channel has equalupstream and downstream capacities, and that the transmit spectrum isspectrally compatible (that is, determined with regard to spectralcompatibility) with the one or more other communications channels.

In one embodiment of the present invention, the transmit spectrum ispreferably determined so that it satisfies a predetermined average powerconstraint for the communications channel. Note that if the channelcapacities depend on the transmit spectra of other lines carrying thesame service, for example in the case of self interference, then thewater filling technique may be carried out as described in reference[16]. If the channel capacity depends on channel noise and/ordifferent-service interference, then the classical water-fillingtechnique is used, as described in [14]. The transmit spectrum ispreferably determined dynamically so that it may be optimized inresponse to changing interference conditions or a changing channeltransfer function.

In another embodiment of the invention, the transmit spectrum isdetermined so that it satisfies both a predetermined average powerconstraint and a predetermined peak power constraint for thecommunications channel, and may be determined using a peak constrainedwater-filling technique. Note that if the channel capacities depend onthe transmit spectra of other lines carrying the same service, forexample in the case of self interference, then the peak constrainedwater filling technique may be carried out as described in section 4.8.3(which presents a modification of the technique discussed in [16]). Ifthe channel capacity depends on channel noise and/or different-serviceinterference, then the peak constrained water-filling technique is used,as described in section 4.8.2. The transmit spectrum is preferablydetermined dynamically so that it may be optimized in response tochanging interference conditions or a changing channel transferfunction.

In another embodiment of the invention, the transmit spectrum isdetermined so that it satisfies only a predetermined peak powerconstraint for the communications channel, and may be determined using apeak constrained water-filling technique.

Dynamical Determination of Transmit Masks

In a preferred embodiment of the invention, steps 322 and 324 ofdetermining the interference characteristics and of determining thetransmit spectrum are performed more than once so that the transmitspectrum is modified appropriately as the interference characteristicschange in time. These steps 322 and 324 may be performed each time adata transfer is initiated. Or, if step 330 of transferring data occursrepeatedly at regular or irregular intervals in time, then steps 322 and324 of determining the interference characteristics and of determiningthe transmit spectrum are preferably performed prior to each occurrenceof transferring data in step 330. In one embodiment of the invention, anew transfer function or a new set of interference characteristics maybe determined during a data transfer and used to calculate a newtransmit spectrum. The new transmit spectrum may then be used in asubsequent portion of the data transfer. These measures of dynamicallydetermining the transfer function enhance the data transfer by allowingthe transfer function to adapt as the characteristics of thecommunications channel change in time.

Orthogonality for Upstream/Downstream Separation and Multi-lineSeparation

In one embodiment of the present invention, the transmit spectrum isdetermined so that it specifies a pair of complementary spectra: one fortransmission in each of the two directions on the communicationschannel. These two spectra may be called the “upstream transmitspectrum” and the “downstream transmit spectrum.” For example, in thecase where the channel provides communication between home 10 and CO 14,the transmit spectrum used in transmission from home 10 may bedesignated the upstream transmit spectrum, while the transmit spectrumused in transmission from CO 14 may be designated the downstreamtransmit spectrum. Similarly, in other cases, such as a well-logging ora multiple-radio-transmitter embodiment, “upstream” and “downstream”indicate opposite directions of transmission as desired.

The upstream and downstream transmit spectra may include one or moreregions of the spectrum that use FDS signaling. In these regions, theupstream and downstream transmit spectra are orthogonal with respect toeach other. In a preferred embodiment of the present invention, thisduplexing orthogonality is achieved by choosing two non-overlappingfrequency subregions in the FDS region, using one of the subregions forupstream signaling, and using the other subregion for downstreamsignaling. More generally, the FDS region may be constructed by choosingtwo non-overlapping sets of frequency subregions in the FDS region,using one of the sets for upstream signaling, and using the other setfor downstream signaling. In another embodiment of the invention theduplexing orthogonality is achieved by using code division signaling(CDS) to separate the upstream and downstream signals in the “FDS”region. In this embodiment, one access code is used in upstreamsignaling, and a second, orthogonal, access code is used in downstreamsignaling.

It is noted that there is an additional benefit to these transmitspectra with one or more regions of FDS signaling: as would beappreciated by one skilled in the art of communications electronics,using regions of orthogonally separated upstream and downstreamsignaling may reduce the overhead of echo cancellation.

The method indicated in FIG. 13 and FIG. 14 may be used in communicatingdata on a communications channel in a situation where one or more of theother communications channels carries the same type of service as thecommunications channel. Under such a condition, step 322 of determininginterference characteristics preferably includes determining an amountof self interference into the communications from the other same-servicecommunications channels. Step 324 of determining the transmit spectrummay then include examining the channel transfer function and the amountof self interference. The transmit spectrum is then preferablydetermined in step 324 in response to the channel transfer function andthe amount of self interference.

In another embodiment of the present invention, the transmit spectrum isdetermined so that it specifies a number M of complementary spectra: onefor transmission on each of M channels in a subset of the one or more ofthe other communications channels that carry the same type of service.These M transmit spectra may include one or more regions of the spectrumthat use MFDS signaling. In these regions, the M transmit spectra areorthogonal with respect to each other. In one embodiment of the presentinvention, this multi-line orthogonality is achieved by choosing Mnon-overlapping frequency subregions in the MFDS region, and using oneof the subregions for transmission on each of the M lines. Moregenerally, the MFDS region may be constructed by choosing Mnon-overlapping sets of frequency subregions in the MFDS region, andusing one of the sets for transmission on each of the M channels. Inanother embodiment of the invention, the multi-line orthogonality isachieved by using multi-line code division signaling (multi-line CDS) inthe “MFDS” region. In this embodiment, different orthogonal access codesare used on each of the M channels.

In another embodiment of the present invention, the transmit spectrum isdetermined so that it specifies a number M′ (>M) of complementaryspectra: one for transmission on each of M channels in the subset ofsame-service channels, and additional spectra to provide orthogonalduplex separation on one or more of the M channels. These M′ transmitspectra may include one or more regions of the spectrum that use FDSsignaling as well as one or more regions of the spectrum that use MFDSsignaling.

In a preferred embodiment of the invention, determining the amount ofself interference comprises determining (1) a self-NEXT transferfunction and (2) a self-FEXT transfer function that describe thecoupling from near-end and far-end transmitters, respectively, on theother same-service communications channels. In this preferred embodimentdetermining the interference characteristics in step 322 furthercomprises determining an amount of uncorrelated interference arisingfrom factors such as additive Gaussian noise (AGN) and crosstalk fromone or more different-service channels, which carry a type of servicedifferent than the service on the channel of interest, among the one ormore other channels. The transmit spectrum is then determined inresponse to the channel transfer function, an average power constraintor requirement for the channel, the self-NEXT and the self-FEXT transferfunctions, and the amount of uncorrelated interference. In regions ofthe communications spectrum where the self interference is substantiallylow, the transmit spectrum is determined to be an EQPSD transmitspectrum. In regions where the self-NEXT interference is substantiallyhigh and the self-FEXT interference is not substantially high, thetransmit spectrum is determined to be an FDS transmit spectrum. And inregions where the self-FEXT interference is substantially high, thetransmit spectrum is determined to be an MFDS transmit spectrum. Somespecific examples of techniques for determining regions of EQPSD, FDS,and MFDS signaling are presented below.

In other embodiments of the invention, determining the interferencecharacteristics in step 322 includes determining some but not all of theself-NEXT and the self-FEXT transfer functions, and the amount ofuncorrelated interference and the average power constraint orrequirement for the channel may or may not be determined. The transmitspectrum is then determined in response to the channel transfer functionand the determined quantities, and is preferably optimized in responseto these quantities.

In one embodiment of the present invention, the transmit spectrum isdetermined in response to one or more characteristics of thecommunications channel and the sources of noise or crosstalk.Determining these characteristics comprises steps such as determiningthe channel transfer function, determining the self-NEXT transferfunction, and determining the self-FEXT transfer function. In onepreferred embodiment, the transmit spectrum is determined in response tothe power-transfer characteristics of the communications channel, sodetermining these characteristics preferably comprises determining onlythe squared modulus of the mathematical transfer functions. Thus, inthis preferred embodiment, determining the channel transfer functionmeans determining the function H(f)≡|H_(C)(f)|², determining theself-NEXT transfer function means determining the functionX(f)≡|H_(N)(f)|², and determining the self-FEXT transfer function meansdetermining the function F(f)≡|H_(F(f)|) ². In this preferredembodiment, the phases of the transfer functions H_(C)(f), H_(N)(f), andH_(F)(f) may or may not be determined in addition to their squaredmodulii. In the case where a distinction is to be made between thevarious functions for different lines, the subscript i is used toindicate the different lines (as in H_(i)(f), X_(i)(f), and F_(i)(f))with channel number i=1 being the channel for which the transmitspectrum is being determined.

Another characteristic of the communications channel and the sources ofnoise or crosstalk is the signal to noise ratio G_(i)(f). In the casewhere a distinction is to be made between different lines, G_(i)(f)indicates, at a frequency f; the ratio of the signal (specifically, thesignal power spectral density at f) in channel number i to the noise(specifically, the noise spectral density at f) in channel number 1.Here, channel number i=1 is the channel for which the transmit spectrumis being determined, and channel number i (for i>1) is another channelthat carries the same type of service as channel number 1, and which mayprovide interference into channel number 1.

Method for Determining Transmission Characteristics with FrequencyBinning

Another embodiment of the present invention comprises a method fordetermining a transmit spectrum for use in communicating data on acommunications channel, preferably by determining signaling techniquesin one or more frequency bins in the available frequency band of thecommunications channel. This method is outlined in the flowchart of FIG.15. This method may be used in communicating data on a communicationschannel when the communications channel is subject to interference fromone or more other communications channels, some of which carry the sametype of service as the communications channel of interest. Additionally,some of the other communications channels may carry different types ofservice than the communications channel of interest.

The first steps in this method comprise determining a channel transferfunction of the communications channel 410. An amount of selfinterference 420 into the communications channel from the othercommunications channels carrying the same type of service is determinedin step 420. An additional amount of uncorrelated interference ispreferably determined in step 425. In step 430, the transfer functionand the amount of self interference are examined, preferably along withthe amount of uncorrelated interference. In step 440 a transmit spectrumfor the channel is determined based on the examining.

In a preferred embodiment of the method, the transmit spectrum isdetermined in step 440 so that different signaling techniques may beused in different frequency ranges in the communications band. Thesefrequency ranges, or frequency bins, are non-overlapping ranges of thefrequency spectrum, preferably with uniform frequency widths, andpreferably chosen so that they cover the communications band. In otherembodiments of the present invention, the frequency bins havenon-uniform widths or do not cover the entire communications band.

In this embodiment of the invention, the transmit spectrum operates tospecify an amount of transmission power used in each frequency bin forat least one direction of communication on at least one communicationschannel. The amount of transmission power in each bin is preferablydetermined by a water-filling technique or a peak constrainedwater-filling technique.

In one embodiment of the invention, in a given frequency bin thetransmit spectrum specifies EQPSD signaling if the amount of selfinterference is substantially low in that bin, and FDS signaling if theamount of self interference is substantially high in that bin.

In one embodiment of the invention, in a given frequency bin thetransmit spectrum specifies MFDS signaling if the amount of self-FEXTinterference is substantially high in that bin. Otherwise, the transmitspectrum specifies EQPSD signaling if the amount of self-NEXTinterference is substantially low in that bin, and FDS signaling if theamount of self-NEXT interference is substantially high in that bin.

Under certain conditions, this method of the present invention maydetermine a transmit spectrum that includes one or more regions ofneighboring bins using FDS signaling. In one embodiment of the presentinvention, the step of determining a transmit spectrum comprisesdetermining a discrete FDS transmit spectrum in such regions ofneighboring FDS bins. In the discrete FDS transmit spectrum, each binhas two subregions; one is used for transmission in the upstreamdirection, and one for transmission in the downstream direction. Inanother embodiment of the present invention, the step of determining atransmit spectrum comprises determining a contiguous FDS transmitspectrum in such regions of neighboring FDS bins. In the contiguous FDStransmit spectrum, the neighboring frequency bins are grouped into twosets of neighboring bins, one of the sets is used for transmission inthe upstream direction, and the other set is used for transmission inthe downstream direction. In one embodiment of the invention, the twosets of neighboring bins are chosen so that the contiguous FDS transmitspectrum provides equal upstream and downstream signaling capacities.Alternatively, the two sets of neighboring bins may be chosen so thatthe contiguous FDS transmit spectrum provides equal upstream anddownstream average power. In a preferred embodiment, the two sets ofneighboring bins are chosen so that the contiguous FDS transmit spectrumprovides equal upstream and downstream signaling capacities and equalupstream and downstream average powers.

Similarly, under certain conditions, this method of the presentinvention may determine a transmit spectrum that includes one or moreregions of neighboring bins using MFDS signaling. In one embodiment ofthe present invention, the step of determining a transmit spectrumcomprises determining a discrete MFDS transmit spectrum in such regionsof neighboring MFDS bins. In the discrete MFDS transmit spectrum, eachbin has M subregions. Each of the M subregions is used forbi-directional transmission on one of the M same-service channels. Inanother embodiment of the present invention, the step of determining atransmit spectrum comprises determining a contiguous MFDS transmitspectrum in such regions of neighboring MFDS bins. In the contiguousMFDS transmit spectrum, the neighboring bins are grouped into M sets ofneighboring bins. Each of the M sets of frequency bins is used forbi-directional transmission on one of the M same-service channels. The Msets of neighboring bins are preferably chosen so that the contiguousMFDS transmit spectrum provides equal signaling capacities on the Mchannels.

If the channel transfer function and the interference characteristicsare substantially monotonic in frequency, the determination of whichfrequency bins use a particular type of signaling may be simplified bydetermining frequency values at which the different types ofinterference become substantially large or substantially small. Thus, inone embodiment of the present invention, determining the transmitspectrum in step 440 includes one or more steps of identifying“transition bins” that mark the endpoints (in the frequency spectrum) ofdifferent types of signaling techniques. These transitions bins may berapidly identified by searching for bins in which certain characteristicquantities meet particular predetermined criteria. These searches, whichare preferably implemented as binary searches, may be carried out in thestep 430 of examining the channel transfer function and interference.The following list is a sample of transition bins that may beidentified.

-   -   M_(E): for bins with center frequencies < or ≦ the center        frequency of M_(E), EQPSD signaling is used.    -   M_(F): for bins with center frequencies > or ≧ the center        frequency of M_(F), FDS signaling is used.    -   M_(E2F): for bins with center frequencies < or ≦ the center        frequency of M_(E2F), EQPSD signaling is used, and FDS signaling        is used in higher-frequency bins. In other words, M_(E2F)        indicates a transition from EQPSD signaling to FDS signaling.    -   M_(E2MFDS): indicates a transition from EQPSD signaling to MFDS        signaling.    -   M_(MFDS2FDS): indicates a transition from MFDS signaling to FDS        signaling.    -   M_(E2MFDS): indicates a transition from EQPSD signaling to FDS        signaling.

Similarly, transition frequencies may be defined for particularfrequencies that mark transitions from one form of signaling to another.For example, f_(E2F) represents a transition frequency where EQPSD isused in a region with frequency less than f_(E2F) and where FDSsignaling is used in a region with frequency less than f_(E2F).

4 New, Optimized Signaling Techniques

The proposed techniques combine a number of ideas into one signalingsystem that optimizes its performance given many different possiblecombinations of interferers. These ideas include:

-   -   1. Given expressions for the crosstalk from other services        (DSIN-NEXT and DSIN-FEXT) into an xDSL channel and channel noise        (AGN), our scheme computes the optimal distribution of power        across frequency that maximizes the capacity (see Section 4.4).        This distribution uses the same transmit spectrum (EQPSD        signaling) in both upstream and downstream directions.    -   2. Given expressions for the self-NEXT and self-FEXT crosstalk        in an xDSL channel along with interference from other services        (DSIN-NEXT and DSIN-FEXT) and channel noise (AGN), our scheme        computes the optimal distribution of power across frequency that        maximizes the capacity. This distribution involves equal PSD        (EQPSD) signaling in frequency bands with low self-interference,        orthogonal signaling (FDS) in frequency bands where self-NEXT        dominates other interference sources (Section 4.5), and        orthogonal signaling (multi-line FDS introduced in Section 4.3)        in frequency bands where self-FEXT is high (Section 4.6).    -   3. Given different channel, noise, and interference        characteristics between lines, our scheme chooses the optimal        signaling strategy (EQPSD, FDS or multi-line FDS) in each        frequency bin (see Section 4.7) to maximize the channel        capacity.    -   4. Given an additional peak-power constraint in frequency, our        scheme computes the optimal transmit spectra that maximize the        capacity and choose the optimal joint signaling strategy (EQPSD,        FDS and multi-line FDS) for a given channel, noise and        interference characteristics (see Sections 4.8 and 4.9).    -   5. We present optimal and near-optimal signaling strategies in        case of non-monotonic channel, self-NEXT and self-FEXT transfer        functions (see Section 4.10 on bridged taps).

We will present the above ideas in the following sections in the contextof a generic xDSL line carrying symmetric-data rate services like HDSL2,“GDSL”, and “VDSL2” services. Note that the techniques developed herecan be applied to a more general communications channel withinterference characteristics characterized by self-interference anddifferent-service interference models. Further, we can extend this workto apply to channels that support asymmetric data rates (different ineach direction) (see Section 4.11), for e.g., ADSL, and VDSL. We canfollow a similar approach of binning in frequency and then analyzing thesignaling strategy in each bin. In the asymmetrical data-rate case, theratio of the average power between upstream and downstream directionsneeds to be known.

We will present background material and our assumptions in Section 4.1.In Section 4.2 we give details about the interference models and thesimulation conditions. Section 4.3 looks at the various signalingschemes we will employ. We will present the optimal transmit spectrumusing EQPSD signaling in Section 4.4 in the presence of onlydifferent-service interference and AGN. Sections 4.5 and 4.6 detail thenew signaling strategies to obtain an optimal and/or suboptimal transmitspectrum in the presence of self-interference, different-serviceinterference and AGN. Section 4.7 derives some results applicable whenneighboring lines vary in channel, noise and interferencecharacteristics. Sections 4.8, and 4.9 present optimal transmit spectraunder additional peak-power constraint in frequency. We present optimaland near-optimal signaling schemes for non-monotonic channel, self-NEXT,and self-FEXT transfer functions in Section 4.10. We discuss optimalsignaling for asymmetrical data-rate channels in Section 4.11. Finally,Section 4.12 presents several new ideas, extending the results presentedhere.

Note: All the transmit spectra are optimal (i.e., yield the maximumpossible bit rates or performance margins) given the assumptions inSection 4.1 (see Sections 4.4.2, 4.5.3, and 4.6.3 for additionalassumptions) and that one of the specific joint signaling strategies isemployed over the channel (see Sections 4.4, 4.5, and 4.6).

4.1 Assumptions, Notation, and Background

We present background material and some of the standard assumptions madefor simulations. These assumptions apply throughout the document unlessnoted otherwise.

-   -   1. Channel noise can be modeled as additive Gaussian noise (AGN)        [13].    -   2. Interference from other services (DSIN-NEXT and DSIN-FEXT)        can be modeled as additive colored Gaussian noise [13].    -   3. We assume the channel can be characterized as a LTI (linear        time invariant) system. We divide the transmission bandwidth B        of the channel into narrow frequency bins of width W (Hz) each        and we assume that the channel, noise and the crosstalk        characteristics vary slowly enough with frequency that they can        be approximated to be constant over each bin (For a given degree        of approximation, the faster these characteristics vary, the        more narrow the bins must be. By letting the number of bins K→∞,        we can approximate any frequency characteristic with arbitrary        precision).¹ We use the following notation for line i on the        channel transfer function [10] $\begin{matrix}        {{{H_{C}(f)}}^{2} = \left\{ \begin{matrix}        H_{i,k} & {{{{if}\quad{{f - f_{k}}}} \leq \frac{W}{2}},} \\        0 & {{otherwise},}        \end{matrix} \right.} & (1)        \end{matrix}$    -    self-NEXT transfer function [8] $\begin{matrix}        {{{H_{N}(f)}}^{2} = \left\{ \begin{matrix}        X_{i,k} & {{{{if}\quad{{f - f_{k}}}} \leq \frac{W}{2}},} \\        0 & {{otherwise},}        \end{matrix} \right.} & (2)        \end{matrix}$    -    and self-FEXT transfer function [9] $\begin{matrix}        {{{H_{F}(f)}}^{2} = \left\{ \begin{matrix}        F_{i,k} & {{{{if}\quad{{f - f_{k}}}} \leq \frac{W}{2}},} \\        0 & {{otherwise}.}        \end{matrix} \right.} & (3)        \end{matrix}$    -    Here f_(k) are the center frequencies (see FIGS. 16 and 17) of        the K subchannels (bins) with index kε{1, . . . , K}. We will        employ these assumptions in Sections 4.5.4, 4.6.6, 4.6.8 and        4.7.1. The DSIN-NEXT and DSIN-FEXT transfer functions are also        assumed to vary slowly enough that they can be similarly        approximated by a constant value in each frequency bin.

¹We divide the channel into narrow frequency bins (or subchannels) forour analysis only. This does not necessarily mean that we need to useDMT as the modulation scheme.

-   -    Note that the concept of dividing a transfer function in        frequency bins is very general and can include nonuniform bins        of varying widths or all bins of arbitrary width (i.e., the bins        need not be necessarily narrow).    -   4. Echo cancellation is good enough that we can ignore crosstalk        from T_(i) ^(o) into R_(i) ^(ō). We can relax this assumption in        some cases where spectral regions employ FDS signaling (see        Sections 4.5, 4.6, 4.7, 4.9, and 4.10).    -   5. All sources of DSIN-NEXT can be lumped into one PSD DS_(N)(f)        and all sources of DSIN-FEXT can be lumped into one PSD        DS_(F)(f).    -   6. AU sources of self-NEXT can be added to form one overall        self-NEXT source.    -   7. All sources of self-FEXT can be added to form one overall        self-FEXT source.    -   8. Spectral optimization is done under the average input power        constraint, i.e., the average input power is limited to P_(max)        (Watts).    -   9. The PSDs of the upstream and downstream transmission        directions can be written using the notation introduced in        Section 1.3.2. There are M interfering lines carrying the same        service with index iε{1, . . . , M}. Denote the direction of        transmission with index oε{u, d}, with u=upstream (to CO) and        d=downstream (from CO). Denote the upstream and downstream PSDs        on line i as:        -   S_(i) ^(u)(f): PSD on twisted pair i in upstream direction            u.        -   S_(i) ^(d)(f): PSD on twisted pair i in downstream direction            d.    -    Further, we denote the upstream and downstream PSD on line i in        a generic frequency bin (or subchannel) k as:        -   s_(i) ^(u)(f): PSD on twisted pair i in upstream direction            u.        -   s_(i) ^(d)(f): PSD on twisted pair i in downstream direction            d.    -    Note: When we refer to s_(i) ^(o)(f) we mean PSD on twisted        pair i in a generic bin, demodulated to baseband (fε[−W, W]) for        ease of notation. When we refer to s^(o)(f) we mean PSD on a        generic twisted pair in a generic bin, demodulated to baseband        (fε[−W, W]) for ease of notation.    -   10. We assume a monotone decreasing channel transfer function.        However, in case the channel transfer function is non-monotonic        (e.g., in the case of bridged taps on the line), our        optimization techniques can be applied in each individual bin        independently. This scenario makes the power distribution        problem more difficult however (see Section 4.10).    -   11. We assume we desire equal channel capacities in upstream and        downstream directions (except when the channel, noise, and        interference characteristics between lines vary as in Section        4.7).        4.2 Interference Models and Simulation Conditions

The interference models for different services have been obtained fromAnnex B of T1.413-1995 ([9], the ADSL standard), with exceptions as inT1E1.4/97-237 [7]. The NEXT coupling model is 2-piece Unger model as inT1E1.4/95-127 [8]. BER was fixed at 10⁻⁷. Our optimal case results weresimulated using Discrete Multitone Technology (DMT) and were comparedwith that of MONET-PAM [1]. MONET-PAM uses Decision Feedback Equalizers(DFE) [20] in the receivers along with multi-level pulse amplitudemodulation (PAM) scheme. The margin calculations for DFE margins weredone per T1E1.4/97-180R1 [11], Section 5.4.2.2.1.1. AGN of power −140dBm/Hz was assumed in both cases. MONET-PAM uses PAM with 3 bits/symboland a baud rate of fbaud=517.33 ksymbols/s. The actual upstream anddownstream power spectra can be obtained from [1]. MONET-PAM spectra islinearly interpolated from 2×1552/3 Hz sampled data. The PAMline-transformer hpf corner, that is, the start frequency is assumed tobe at 1 kHz. A 500 Hz rectangular-rule integration is carried out tocompute margins. The required DFE SNR margin for 10⁻⁷ BER is 27.7 dB.

To implement our optimal signaling scheme, we used DMT with startfrequency 1 kHz and sampling frequency of 1 MHz. This gives us abandwidth of 500 kHz and 250 carriers with carrier spacing of 2 kHz. Nocyclic prefix (used to combat intersymbol interference (ISI)) wasassumed, so the DMT symbol rate is same as the carrier spacing equal to2 kHz. However, the scheme can easily be implemented by accounting foran appropriate cyclic prefix. The addition of cyclic prefix lowers thesymbol rate and hence lowers the transmission rate. No limit was imposedon the maximum number of bits per carrier (this is often done forsimulations). Even with a 15 bits/carrier limit, the results should notchange very much, as some of the test runs show.

4.3 Signaling Schemes

The joint signaling techniques used in the overall optimized signalingschemes use one of the basic signaling schemes (see FIG. 18) indifferent frequency bins depending on the crosstalk and noisecombination in those bins.

FIG. 18 illustrates the three signaling schemes: EQPSD, FDS andmulti-line FDS (in the case of three lines).² The Figure shows infrequency bin k the PSDs for each case (recall the notation introducedin Section 4.1, Item 9):

²The signaling schemes EQPSD, FDS, and multi-line FDS work in generalfor M lines.

-   -   When crosstalk and noise are not significant in a frequency bin,        EQPSD signaling is preferred as it achieves higher bit rate than        the other two orthogonal signaling schemes (see Section 4.5.5).        In EQPSD signaling, the upstream and downstream PSDs are the        same (s_(i) ^(u)(f)=s_(i) ^(d)(f)).    -   When self-NEXT is high and self-FEXT is low in a bin and there        are a large number of neighboring lines carrying the same        service together, FDS signaling yields the highest bit rates by        eliminating self-NEXT (we prove this in Section 4.5.5). In FDS        signaling, each frequency bin is further divided into two        halves, with all the upstream PSDs being same for all the lines        and all the downstream PSDs being same for all the lines (s_(i)        ^(u)(f)⊥s_(i) ^(d)(f)). This type of orthogonal signaling        completely eliminates self-NEXT but does not combat self-FEXT.    -   In frequency bins where self-FEXT is high, using FDS is not        sufficient since self-FEXT still exists. In this case, doing        multi-line FDS eliminates self-FEXT as well as self-NEXT and        this achieves the highest bit rates when there are only a few        lines and self-FEXT is high and dominant over self-NEXT (we        prove this in Section 4.6). In multi-line FDS signaling each        line gets a separate frequency slot (W/M for M lines carrying        the same service) in each bin and the upstream and downstream        PSDs for each line are the same (s_(i) ^(o)(f)⊥s_(j) ^(o)(f)        ∀j≠i, oε{u, d}).

We will see in future sections the exact relationships that allow us todetermine which scheme is optimal given an interference and noisecombination.

4.4 Optimization: Interference from Other Services (DSIN-NEXT andDSIN-FEXT)—Solution: EQPSD Signaling

In this scenario, each xDSL line experiences no self-interference (FIG.19 with neither self-NEXT nor self-TEXT). There is only DSIN-NEXT andDSIN-FEXT from other neighboring services such as T1, ADSL, HDSL, etc.,in addition to AGN. The solution is well known, but will be useful laterin the development of the subsequent novel (Sections 4.5, 4.6, 4.7, and4.12) signaling schemes.

4.4.1 Problem Statement

Maximize the capacity of an xDSL line in the presence of AGN andinterference (DSIN-NEXT and DSIN-FEXT) from other services under twoconstraints:

-   -   1. The average xDSL input power in one direction of transmission        must be limited to P_(max) (Watts).    -   2. Equal capacity in both directions (upstream and downstream)        for xDSL.        Do this by designing the distribution of energy over frequency        (the transmit spectrum) of the xDSL transmission.

4.4.2 Additional Assumption

We add the following assumption to the ones in Section 4.1 for thiscase:

-   -   12. Both directions (upstream and downstream) of transmission        experience the same channel noise (AGN) and different service        interference (DSIN-NEXT and DSIN-FEXT).

4.4.3 Solution

Consider a line (line 1) carrying xDSL service. Line 1 experiencesinterference from other neighboring services (DSIN-NEXT and DSIN-FEXT)and channel noise N_(o)(f) (AGN) but no self-NEXT or self-FEXT (see FIG.19).

The DSIN-NEXT and DSIN-FEXT interference can be modeled as coloredGaussian noise for calculating capacity [13]. Recall that DS_(N)(f) isthe PSD of the combined DSIN-NEXT and let DS_(F)(f) is the PSD of thecombined DSIN-FEXT. Let S^(u)(f) and S^(d)(f) denote the PSDs of line 1upstream (u) direction and downstream (d) direction transmitted signalsrespectively. Further, let C^(u) and C^(d) denote the upstream anddownstream direction capacities of line 1 respectively. Let H_(C)(f)denote the channel transfer function of line 1. The twisted pair channelis treated as a Gaussian channel with colored Gaussian noise. In thiscase the channel capacity (in bps) is given by [14] $\begin{matrix}{{C^{u} = {\sup\limits_{S^{u}{(f)}}{\int_{0}^{\infty}{{\log_{2}\left\lbrack {1 + \frac{{{H_{C}(f)}}^{2}{S^{u}(f)}}{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)}}} \right\rbrack}{\mathbb{d}f}}}}}{and}} & (4) \\{C^{d} = {\sup\limits_{S^{d}{(f)}}{\int_{0}^{\infty}{{\log_{2}\left\lbrack {1 + \frac{{{H_{C}(f)}}^{2}{S^{d}(f)}}{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)}}} \right\rbrack}{{\mathbb{d}f}.}}}}} & (5)\end{matrix}$The supremum is taken over all possible S^(u)(f) and S^(d)(f) satisfyingS ^(u)(f)≧0∀f, S ^(d)(f)≧0∀f,and the average power constraints for the two directions $\begin{matrix}{{{2{\int_{0}^{\infty}{{S^{u}(f)}{\mathbb{d}f}}}} \leq P_{m\quad{ax}}},{{{and}\quad 2{\int_{0}^{\infty}{{S^{d}(f)}{\mathbb{d}f}}}} \leq {P_{m\quad{ax}}.}}} & (6)\end{matrix}$It is sufficient to find the optimal S^(u)(f) which gives C^(u), sincesetting S^(d)(f)=S^(u)(f) ∀f, gives the capacity C^(d)=C^(u) as seenfrom (4) and (5). Thus, the optimal upstream and downstream channelcapacities are equal (C^(u)=C^(d)).

The optimal power distribution in this case is obtained by the classical“water-filling” technique [16]. The optimal S^(u)(f) is given by$\begin{matrix}{{S_{opt}^{u}(f)} = \left\{ {\begin{matrix}{\lambda - \frac{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)}}{{{H_{C}(f)}}^{2}}} & {{{for}\quad f} \in E} \\0 & {otherwise}\end{matrix},} \right.} & (7)\end{matrix}$with λ a Lagrange multiplier and E the spectral region where S^(u)(f)≧0.We vary the value of λ such that S_(opt) ^(u)(f) satisfies with equalitythe average power constraint in (6). The equality is satisfied for asingle value of λ giving us a unique optimal PSD S_(opt) ^(u)(f).Plugging the optimal PSD S_(opt) ^(u)(f) in (4) yields the capacityC^(u) under the average power constraint. This procedure yields a uniqueoptimal transmit spectrum S_(opt) ^(u)(f) [14].

Keynote: S^(d)(f)=S^(u)(f) ∀f—EQPSD signaling.

FIG. 20 gives a flowchart to obtain the optimal transmit spectrum usingonly EQPSD signaling in the presence of DSIN-NEXT, DSIN-FEXT and AGN. Ituses the classic water-filling solution to obtain the transmit spectrum.The novelty is in applying this to xDSL scenario to achieve a dynamictransmit spectrum (different for each interference type).

The channel capacities can be calculated separately for each directionof transmission in case of nonuniform interference between the twodirections, i.e., when the additional assumption in Section 4.4.2 doesnot hold. The transmit spectra in general will be different(S^(d)(f)≠S^(u)(f)) for this case, but will still occupy the samebandwidth.

4.4.4 Examples

In this Section, we present some examples for the HDSL2 service. Anaverage input power (P_(max)) of 20 dBm and a fixed bit rate of 1.552Mbps was used for all simulations. The performance margin was measuredin each simulation and the comparison with other static transmit spectra(obtained from static PSD masks) proposed is presented in Section4.5.11. FIG. 21 shows the optimal upstream and downstream transmitspectrum for HDSL2 in the presence of DSIN-NEXT from 49 HDSL interferersand AGN (−140 dBm/Hz). Note the deep null in the transmit spectrum fromapproximately 80 to 255 kHz. This results from “water-filling”—the peakof the first main lobe of HDSL lies in the vicinity of 80 to 255 kHz.

FIG. 22 shows the optimal upstream and downstream transmit spectrum forHDSL2 in the presence of DSIN-NEXT from 25 T1 interferers and AGN (−140dBm/Hz).

The optimal transmit spectra for the two cases are significantlydifferent, evidence of the fact that the optimal transmit spectra willchange depending on the nature of the interference.

Summary: Recall the discussion on static PSD masks of Section 3.1. Wehave seen that the optimal transmit spectrum vanes significantly withthe interference combination. The water-filling solution yields a uniquetransmit spectrum for each interference combination [14]. The optimaltransmit spectrum adapts to minimize the effect of the interferencecombination. The optimal transmit spectra for upstream and downstreamdirection are the same (EQPSD signaling) and thus, employ the sameaverage power in each direction.

4.5 Optimization: Interference from Other Services (DSIN-NEXT andDSIN-FEXT) Plus Self-interference (Self-NEXT and LowSelf-FEXT)—Solution: EQPSD and FDS Signaling

In this scenario each xDSL line experiences self-interference (highself-NEXT and low self-FEXT) in addition to AGN and DSIN-NEXT andDSIN-FEXT from other services (see FIG. 3) in a generic xDSL service.This is the case of interest for HDSL2 service.

4.5.1 Self-NEXT and Self-FEXT Rejection Using Orthogonal Signaling

As we saw in Section 3.2, orthogonal signaling can completely rejectself-NEXT. In addition, FDS gives better spectral compatibility withother services than other orthogonal schemes like CDS or TDS (seeSection 4.5.12 for a proof). Therefore, we choose to use the FDS schemefor orthogonal signaling. Recall the FDS signaling tradeoff: FDSeliminates self-NEXT and therefore increases system capacity; however,FDS also reduces the bandwidth available to each transmitter/receiverpair and therefore decreases system capacity.

To eliminate self-FEXT using orthogonal signaling, we would force eachupstream transmitter T_(i) ^(u) to be orthogonal to all othertransmitters T_(j) ^(u)≠i. Using multi-line FDS, we would separate eachT_(i) ^(u) into different frequency bands. Unfortunately, this wouldreduce the bandwidth available to each transmitter to 1/M the overallchannel bandwidth. In a typical implementation of HDSL2, M will liebetween 1 and 49; hence orthogonal signaling (multi-line FDS) foreliminating self-FEXT is worth the decrease in capacity only whenself-FEXT is very high. We will show later in Section 4.6 thatmulti-line FDS gives gains in capacity when there are only a few numberof interfering lines carrying the same service (M=2 to 4).

In this scenario, we assume self-NEXT dominates self-FEXT and self-FEXTis not very high (see FIG. 17 and [8]), so we will design a system herewith only self-NEXT suppression capability. However, self-FEXT stillfactors into our design in an important way. This is a new, non-trivialextension of the work of [3].

4.5.2 Problem Statement

Maximize the capacity of an xDSL line in the presence of AGN,interference (DSIN-NEXT and DSIN-FEXT) from other services, andself-NEXT and self-FEXT under two constraints:

-   -   1. The average xDSL input power in each direction of        transmission must be limited to P_(max) (Watts), and    -   2. Equal capacity in both directions (upstream and downstream)        for xDSL.        Do this by designing the distribution of energy over frequency        (the transmit spectrum) of the upstream and downstream xDSL        transmissions.

4.5.3 Additional Assumptions

We add the following assumptions to the ones in Section 4.1 for thiscase:

-   -   12. The level of self-FEXT is low enough in all bins that it is        not necessary to use orthogonal signaling between different        transmitter/receiver pairs operating in the same direction (see        Section 4.5.1).    -   13. All the M lines considered are assumed to have the same        channel and noise characteristics and face the same interference        combination (interference combination refers to combination of        different interfering services) in both transmission directions        (upstream and downstream). We will develop some results in        Section 4.7 for when this does not hold true. Thus, we assume        that the upstream PSDs of all lines are the same (S^(u)(f)) and        the downstream PSDs of all lines are the same (S^(d)(f)). That        is,        S ^(u)(f)=S _(i) ^(u)(f), iε{1, . . . , M}        S ^(d)(f)=S _(i) ^(d)(f), iε{1, . . . , M}  (8)    -   14. The coupling transfer functions of NEXT and FEXT        interference are symmetrical between neighboring services. For        example, each line has the same self-NEXT transfer function        H_(N)(f) and self-FEXT transfer function H_(F)(f) for computing        coupling of interference power with any other line. However, we        develop some results in Section 4.7 when there are different        NEXT and FEXT coupling transfer functions between lines.

4.5.4 Signaling Scheme

Since the level of self-NEXT will vary with frequency (recall FIG. 17),it is clear that in high self-NEXT regions of the spectrum, orthogonalsignaling (FDS, for example) might be of use in order to rejectself-NEXT. However, in low self-NEXT regions, the loss of transmissionbandwidth of FDS may outweigh any gain in capacity due to self-NEXTrejection. Therefore, we would like our signaling scheme to be generalenough to encompass both FDS signaling, EQPSD signaling, and thespectrum of choices in between. Our approach is related to that of [3].

Key to our scheme is that the upstream and downstream transmissions usedifferent transmit spectra. All upstream (to CO) transmitters T_(i) ^(u)transmit with the spectrum S^(u)(f) All downstream (from CO)transmitters T_(i) ^(d) transmit with the spectrum S^(d)(f) Implicit inour scheme is the fact that in this case, self-NEXT dominates self-FEXTand self-FEXT is small. If not, it would not be wise to constrain allT_(i) ^(u) to the same transmit PSD.

Our goal is to maximize the upstream capacity (C^(u)) and the downstreamcapacity (C^(d)) given an average total power constraint of P_(max) andthe equal capacity constraint C^(u)=C^(d).

Consider the case of two lines with the same service. Line 1 upstreamcapacity is C^(u) and line 2 downstream capacity is C^(d). Under theGaussian channel assumption, we can write these capacities (in bps) as$\begin{matrix}{{C^{u} = {\sup\limits_{{S^{u}{(f)}},{S^{d}{(f)}}}{\int_{0}^{\infty}{{\log_{2}\left\lbrack {1 + \frac{{{H_{C}(f)}}^{2}{S^{u}(f)}}{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)} + {{{H_{N}(f)}}^{2}{S^{d}(f)}} + {{{H_{F}(f)}}^{2}{S^{u}(f)}}}} \right\rbrack}{\mathbb{d}f}}}}},{and}} & (4) \\{C^{d} = {\sup\limits_{{S^{u}{(f)}},{S^{d}{(f)}}}{\int_{0}^{\infty}{{\log_{2}\left\lbrack {1 + \frac{{{H_{C}(f)}}^{2}{S^{d}(f)}}{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)} + {{{H_{N}(f)}}^{2}{S^{u}(f)}} + {{{H_{F}(f)}}^{2}{S^{d}(f)}}}} \right\rbrack}{{\mathbb{d}f}.}}}}} & (5)\end{matrix}$The supremum is taken over all possible S^(u)(f) and S^(d)(f) satisfyingS ^(u)(f)≧0∀f, S ^(d)(f)≧0∀f,and the average power constraints for the two directions $\begin{matrix}{{{2{\int_{0}^{\infty}{{S^{u}(f)}{\mathbb{d}f}}}} \leq P_{m\quad{ax}}},{{{and}\quad 2{\int_{0}^{\infty}{{S^{d}(f)}{\mathbb{d}f}}}} \leq {P_{m\quad{ax}}.}}} & (11)\end{matrix}$

We can solve for the capacities C^(u) and C^(d) using “water-filling” ifwe impose the restriction of EQPSD, that is S^(u)(f)=S^(d)(f) ∀f.However, this gives low capacities. Therefore, we employ FDS (S^(u)(f)orthogonal to S^(d)(f)) in spectral regions where self-NEXT is largeenough to limit our capacity and EQPSD in the remaining spectrum. Thisgives much improved performance.

To ease our analysis, we divide the channel into several equal bandwidthsubchannels (bins) (see FIG. 16) and continue our design and analysis onone frequency bin k assuming the subchannel frequency responses (1)-(3).Recall that FIG. 17 shows that the channel and self-interferencefrequency responses are smooth and justifies our assuming them flat overnarrow subchannels. For ease of notation, in this Section setH=H _(i,k) , X=X _(i,k) , F=F _(i,k) in (1)-(3),  (12)andN=N _(o)(f _(k))+DS _(N)(f _(k))+DS _(F)(f _(k)),  (13)the noise PSD in bin k. Note that N consists of both AGN plus anyinterference (DSIN-NEXT and DSIN-FEXT) from other services. Let s^(u)(f)denote the PSD in bin k of line 1 upstream direction and s^(d)(f) denotethe PSD in bin k of line 2 downstream direction (recall the notationintroduced in Section 4.1, Item 9). The corresponding capacities of thesubchannel k are denoted by c^(u) and c^(d).

We desire a signaling scheme that includes FDS, EQPSD and allcombinations in between in each frequency bin. Therefore we divide eachbin in half³ and define the upstream and downstream transmit spectra asfollows (see FIG. 23): $\begin{matrix}{{s^{u}(f)} = \left\{ {\begin{matrix}{\alpha\frac{2P_{m}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\left( {1 - \alpha} \right)\frac{2\quad P_{m}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {otherwise}\end{matrix}{and}} \right.} & (14) \\{{s^{d}(f)} = \left\{ \begin{matrix}{\left( {1 - \alpha} \right)\frac{2P_{m}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\alpha\frac{2\quad P_{m}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise}.}\end{matrix} \right.} & (15)\end{matrix}$Here P_(m) is the average power over frequency range [0, W] in bin k and0.5≦α≦1. When α=0.5, s^(u)(f)=s^(d)(f) ∀fε[0, W] (EQPSD signaling); whenα=1, s^(u)(f) and s^(d)(f) are disjoint (FDS signaling). These twoextreme transmit spectra along with other possible spectra (fordifferent values of α) are illustrated in FIG. 23. The PSDs s^(u)(f) ands^(d)(f) are “symmetrical” or power complementary to each other. Thisensures that the upstream and downstream capacities are equal(c^(u)=c^(d)). The factor α controls the power distribution in the bin,and W is the bandwidth of the bin.

³The power split-up in a bin does not necessarily have to be 50% to theleft side of the bin and 50% to the right side of the bin as shown inFIG. 23. In general any 50%—50% power-complementary split-up betweenopposite direction bins will work.

Next, we show that given this setup, the optimal signaling strategy usesonly FDS or EQPSD in each subchannel.

4.5.5 Solution: One Frequency Bin

If we define the achievable rate as $\begin{matrix}{{{R_{A}\left( {{s^{u}(f)},{s^{d}(f)}} \right)} = {\int_{0}^{W}{{\log_{2}\left\lbrack {1 + \frac{{s^{u}(f)}H}{N + {{s^{d}(f)}X} + {{s^{u}(f)}F}}} \right\rbrack}{\mathbb{d}f}}}},} & (16) \\{then} & \quad \\{{c^{u} = {\max\limits_{0.5 \leq \alpha \leq 1}{{R_{A}\left( {{s^{u}(f)},{s^{d}(f)}} \right)}\quad{and}}}}\quad{c^{d} = {\max\limits_{0.5 \leq \alpha \leq 1}{{R_{A}\left( {{s^{d}(f)},{s^{u}(f)}} \right)}.}}}} & (17)\end{matrix}$Due to the power complementarity of s^(u)(f) and s^(d)(f) the channelcapacities c^(u) and c^(d) are equal. Therefore, we will only considerthe upstream capacity c^(u) expression. Further, we will use R_(A) forR_(A)(s^(u)(f), s^(d)(f)) in the remainder of this Section. Substitutingfor the PSDs from (14) and (15) into (16) and using (17) we get thefollowing expression for the upstream capacity $\begin{matrix}{c^{u} = {\frac{W}{2}{\max\limits_{0.5 \leq \alpha \leq 1}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\frac{\alpha\quad 2P_{m}H}{W}}{N + \frac{\left( {1 - \alpha} \right)2P_{m}X}{W} + \frac{\alpha\quad 2P_{m}F}{W}}} \right\rbrack} + {\log_{2}\left\lbrack {1 + \frac{\frac{\left( {1 - \alpha} \right)2P_{m}H}{W}}{N + \frac{{\alpha 2}\quad P_{m}X}{W} + \frac{\left( {1 - \alpha} \right)2\quad P_{m}F}{W}}} \right\rbrack}} \right\}.}}}} & (18)\end{matrix}$Let $G = \frac{2\quad P_{m}}{WN}$denote the SNR in the bin. Then, we can rewrite (18) as $\begin{matrix}{c^{u} = {\max\limits_{0.5 \leq \alpha \leq 1}{\frac{W}{2}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\alpha\quad{GH}}{1 + {\left( {1 - \alpha} \right){GX}} + {\alpha\quad{GF}}}} \right\rbrack} + {\log_{2}\left\lbrack {1 + \frac{\left( {1 - \alpha} \right){GH}}{1 + {\alpha\quad{GX}} + {\left( {1 - \alpha} \right){GF}}}} \right\rbrack}} \right\}.}}}} & (19)\end{matrix}$

Note from (17) and (19) that the expression after the max in (19) is theachievable rate R_(A). Differentiating the achievable rate (R_(A))expression in (19) with respect to α gives us $\begin{matrix}\begin{matrix}{\frac{\partial R_{A}}{\partial\alpha} = {\frac{W}{2\ln\quad 2}\left\{ \left\lbrack {\frac{1 + {\left( {1 - \alpha} \right){GX}} + {\alpha\quad{GF}}}{1 + {\left( {1 - \alpha} \right){GX}} + {\alpha\quad{GF}} + {\alpha\quad{GH}}} \times} \right. \right.}} \\{\left. \frac{{{GH}\left( {1 + {\left( {1 - \alpha} \right){GX}} + {\alpha\quad{GF}}} \right)} - {\alpha\quad{{GH}\left( {{- {GX}} + {GF}} \right)}}}{\left( {1 + {\left( {1 - \alpha} \right){GX}} + {\alpha\quad{GF}}} \right)^{2}} \right\rbrack +} \\{\left\lbrack {\frac{1 + {\alpha\quad{GX}} + {\left( {1 - \alpha} \right){GF}}}{1 + {\alpha\quad{GX}} + {\left( {1 - \alpha} \right){GF}} + {\left( {1 - \alpha} \right){GH}}} \times} \right.} \\\left. \left. \frac{{- {{GH}\left( {1 + {\alpha\quad{GX}} + {\left( {1 - \alpha} \right){GF}}} \right)}} - {\left( {1 - \alpha} \right){{GH}\left( {{GX} - {GF}} \right)}}}{\left( {1 + {\alpha\quad{GX}} + {\left( {1 - \alpha} \right){GF}}} \right)^{2}} \right\rbrack \right\}\end{matrix} & (20) \\{{= {{{G\left( {{2\quad\alpha} - 1} \right)}\left\lbrack {{2\left( {X - F} \right)} + {G\left( {X^{2} - F^{2}} \right)} - {H\left( {1 + {GF}} \right)}} \right\rbrack}L}},} & (21)\end{matrix}$with L>0 ∀αε(0, 1]. Setting the derivative to zero gives us the singlestationary point α=0.5. The achievable rate R_(A) is monotonic in theinterval αε(0.5, 1] (see FIG. 24). If the value α=0.5 corresponds to amaximum, then it is optimal to perform EQPSD signaling in this bin. Ifthe value α=0.5 corresponds to a minimum, then the maximum is achievedby the value α=1, meaning it is optimal to perform FDS signaling in thisbin. No other values of α are an optimal option. See FIG. 25.

The quantity α=0.5 corresponds to a maximum of R_(A) (EQPSD) if and onlyif $\frac{\partial R_{A}}{\partial\alpha} < 0$∀αε(0.5, 1]. For all αε(0.5, 1], the quantity (2α−1) is positive and$\frac{\partial R_{A}}{\partial\alpha}$is negative if and only if (see (21))2(X−F)+G(X ² −F ²)−H(1+GF)<0.This implies thatG(X ² −F ² −HF)<H−2(X−F).Thus, the achievable rate R_(A) is maximum at α=0.5 (EQPSD)$\begin{matrix}{{{{if}\quad X^{2}} - F^{2} - {HF}} < {0\quad{and}\quad G} > \frac{H - {2\left( {X - F} \right)}}{X^{2} - F^{2} - {HF}}} & (22) \\{or} & \quad \\{{{{if}\quad X^{2}} - F^{2} - {HF}} > {0\quad{and}\quad G} < {\frac{H - {2\left( {X - F} \right)}}{X^{2} - F^{2} - {HF}}.}} & (23)\end{matrix}$

In a similar fashion α=0.5 corresponds to a minimum of R_(A) if and onlyif $\frac{\partial R_{A}}{\partial\alpha} > 0$∀αε(0.5, 1]. This implies that α=1 corresponds to a maximum of R_(A)(FDS) since there is only one stationary point in the interval αε[0.5,1] (see FIG. 24). For all αε(0.5, 1],$\frac{\partial R_{A}}{\partial\alpha}$is positive if and only if2(X−F)+G(X ² −F ²)−H(1+GF)>0.This implies thatG(X ² −F ² −HF)>H−2(X−F).Thus, the achievable rate R_(A) is maximum at α=1 (FDS) $\begin{matrix}{{{{if}\quad X^{2}} - F^{2} - {HF}} < {0\quad{and}\quad G} < \frac{H - {2\left( {X - F} \right)}}{X^{2} - F^{2} - {HF}}} & (24) \\{or} & \quad \\{{{{if}\quad X^{2}} - F^{2} - {HF}} > {0\quad{and}\quad G} > {\frac{H - {2\left( {X - F} \right)}}{X^{2} - F^{2} - {HF}}.}} & (25)\end{matrix}$

Thus, we can determine whether the value α=0.5 maximizes or minimizesthe achievable rate by evaluating the above inequalities. If α=0.5corresponds to a maximum of R_(A), then we achieve capacity c^(u) bydoing EQPSD signaling. If α=0.5 corresponds to a minimum of R_(A), thenwe achieve capacity c^(u) by doing FDS signaling. This can be summed intest conditions to determine the signaling nature (FDS or EQPSD) in agiven bin. Using (22) and (24) we can write

If X ² −F ² −HF<0 then $\begin{matrix}{G = {\frac{2P_{m}}{NW}\begin{matrix}\begin{matrix}\begin{matrix}{EQPSD} \\ > \end{matrix} \\ < \end{matrix} \\{FDS}\end{matrix}{\frac{H - {2\left( {X - F} \right)}}{X^{2} - F^{2} - {HF}}.}}} & (26)\end{matrix}$

Also, using (23) and (25) we can write

If X ² −F ² −HF>0 then $\begin{matrix}{G = {\frac{2P_{m}}{NW}\begin{matrix}\begin{matrix}\begin{matrix}{EQPSD} \\ > \end{matrix} \\ < \end{matrix} \\{FDS}\end{matrix}{\frac{H - {2\left( {X - F} \right)}}{X^{2} - F^{2} - {HF}}.}}} & (27)\end{matrix}$

Thus, we can write the upstream capacity c^(u) in a frequency bin k as$\begin{matrix}{c^{u} = \left\{ \begin{matrix}{{W\quad{\log_{2}\left\lbrack {1 + \frac{P_{m}H}{{NW} + {P_{m}\left( {X + F} \right)}}} \right\rbrack}},} & {{{{if}\quad\alpha} = 0.5},} \\{{\frac{W}{2}{\log_{2}\left\lbrack {1 + \frac{P_{m}H}{{N\frac{W}{2}} + {P_{m}F}}} \right\rbrack}},} & {{{if}\quad\alpha} = 1.}\end{matrix} \right.} & (28)\end{matrix}$

Note: Its always optimal to do either EDS or EQPSD signaling; that is,α=0.5 or 1 only. FDS signaling scheme is a subset of the more generalorthogonal signaling concept. However, of all orthogonal signalingschemes, FDS signaling gives the best results in terms of spectralcompatibility under an average power constraint and hence is used here(see proof in Section 4.5.12). In the case of a peak power constraint infrequency, other orthogonal schemes, such as CDS, could be moreappropriate (see Section 4.12.6).

4.5.6 Solution: All Frequency Bins

We saw in Section 4.5.5 how to determine the optimal signaling scheme(FDS or EQPSD) in one frequency bin for the upstream and downstreamdirections. In this Section we will apply the test conditions in (26)and (27) to all the frequency bins to determine the overall optimalsignaling scheme. Further, using “water-filling” (this comprises of theclassical water-filling solution (14] and an optimization technique tocompute capacity in the presence of self-interference [16]) optimize thepower distribution over the bins given the average input power(P_(max)).

We divide the channel into K narrow subchannels of bandwidth W (Hz) each(see FIG. 16). For each subchannel k, we compute the respective channeltransfer function (H_(C)(f_(k)), self-NEXT (H_(N)(f_(k))), self-FEXT(H_(F)(f_(k))), DSIN-NEXT (DS_(N)(f_(k))), DSIN-FEXT (DS_(F)(f_(k))) andAGN (N_(o)(f_(k))). Then, by applying (26) and (27) to each bin k in thegeneric xDSL scenario (with the usual monotonicity assumptions asoutlined in Section 4.1),⁴ we can divide the frequency axis (K bins)into 3 major regions:

-   -   1. The right side of (26)<0 for bins [1, M_(E)]. These bins        employ EQPSD signaling (since power in every bin is ≧0).    -   2. The right side of (27)<0 for bins [M_(F), K]. These bins        employ FDS signaling (since power in every bin is ≧0) and        M_(E)<M_(F).    -   3. The signaling scheme switches from EQPSD to FDS signaling at        some bin M_(E2F), which lies in the range of bins (M_(E),        M_(F)).

⁴When the channel transfer function is non-monotonic (as in the case ofbridged taps) a bin-by-bin approach may be required to achieve theoptimal power distribution (see Section 4.10).

FIG. 26 illustrates the situation of the 3 bins M_(E), M_(F) andM_(E2F). In the next Section we develop an algorithm to find the optimalbin M_(E2F) and the optimal power distribution.

4.5.7 Algorithm for Optimizing the Overall Transmit Spectrum

To find the optimal EQPSD to FDS switch-over bin M_(E2F) and the optimalpower distribution over all bins:

-   -   1. Set up equispaced frequency bins of width W (Hz) over the        transmission bandwidth B of the channel. The bins should be        narrow enough for the assumptions (1)-(3) of Section 4.1 to        hold.    -   2. Estimate the interference (DSIN-NEXT, DSIN-FEXT, self-NEXT        and self-FEXT) and noise (AGN) PSDs. Lump the corresponding        interference PSDs together into one PSD.    -   3. Compute the bins M_(E) and M_(F) using (26) and (27) as        outlined in Section 4.5.6    -   4. Choose an initial estimate of M_(E2F) (M_(E) is a great        start).    -   5. Choose an initial distribution of how much proportion of the        total power (P_(max)) should go in the spectrum to the left of        M_(E2F) and how much should go to the right. Denote these powers        by P_(E) and P_(F)=P_(max)−P_(E) respectively.    -   6. Use water-filling to distribute these powers (P_(E) and        P_(F)) optimally over frequency [14, 16] with EQPSD signaling in        bins [1, M_(E2F)] and FDS signaling in bins [M_(E2F)+1, K].        Compute the subchannel capacity c^(u) in each bin using (28).        Calculate the channel capacity C^(u) by summing all subchannel        capacities.    -   7. Re-estimate the powers P_(E) and P_(F).    -   8. Repeat steps 6 to 7 for a range of powers P_(E) and P_(F) in        search of the maximum channel capacity C^(u). This search is        guaranteed to converge [3].    -   9. Re-estimate the optimal EQPSD to FDS switch-over bin M_(E2F).    -   10. Repeat steps 5 to 9 for a range of bin values for M_(E2F).    -   11. Choose the bin number which yields the highest channel        capacity C^(u) as the true optimal bin M_(E2F) after which the        signaling switches from EQPSD to FDS.        Notes:    -   1. Standard minimization/maximization routines (like fmin in the        software package MATLAB) can be used to search for the optimal        powers P_(E) and P_(F).    -   2. We can use fast algorithms like the Golden Section Search        [19] to find the optimal bin M_(E2F). This routine tries to        bracket the minimum/maximum of the objective function (in this        case capacity) using four function-evaluation points. We start        with a triplet (p, q, r) that brackets the minimum/maximum. We        evaluate the function at a new point xε(q, r) and compare this        value with that at the two extremeties to form a new bracketing        triplet (p, q, x) or (q, x, r) for the minimum/maximum point. We        repeat this bracketing procedure till the distance between the        outer points is tolerably small.    -   4.5.8 Fast, Suboptimal Solution for the EQPSD to FDS Switch-over        Bin

In the estimation of the optimal bin M_(E2F) we have observed inpractice that M_(E2F)≈M_(E), typically within 1 or 2 bins especiallywhen self-interference dominates the total crosstalk (see Section4.5.11). In the case of low AGN and different-service interference thesuboptimal solution is a substantially optimized solution. Thus, withsignificantly less computational effort than the algorithm described inSection 4.5.7, a near-optimal solution can be obtained. Even if a searchis mounted for M_(E2F), we suggest that the search should start at M_(E)(and move to the right).

Algorithm to implement the suboptimal solution:

-   -   1. Perform Steps 1 and 2 of the algorithm of Section 4.5.7.    -   2. Compute the bin M_(E) using (26) as outlined in Section        4.5.6.    -   3. Set the EQPSD to FDS switch-over bin M_(E2F) equal to M_(E).    -   4. Obtain the optimal power distribution and the channel        capacity C^(u) by performing Steps 5 through 8 of the algorithm        in Section 4.5.7.

4.5.9 Flow of the Scheme

Consider a line carrying an xDSL service satisfying the assumptions ofSections 4.1 and 4.5.3. Lines carrying the same xDSL service anddifferent xDSL services interfere with the line under consideration. Wewish to find the optimal transmit spectrum for the xDSL line underconsideration (see problem statement in Section 4.5.2).

-   -   1. Determine the self-NEXT and self-FEXT levels due to other        xDSL lines, bin by bin. These can be determined either through:        -   (a) a worst-case bound of their levels determined by how            many lines of that xDSL service could be at what proximity            to the xDSL line of interest; or        -   (b) an adaptive estimation (training) procedure run when the            modem “turns on.” In this process the CO will evaluate the            actual number of active self-interfering xDSL lines and the            proximity of those lines with the line of interest.    -   2. Determine DSIN-NEXT and DSIN-FEXT levels, bin by bin. These        can be determined either through:        -   (a) a worst-case bound of their levels determined by how            many lines of which kinds of service could be at what            proximity to the xDSL line of interest; or        -   (b) an adaptive estimation (training) procedure run when the            modem “turns on”. In this procedure no signal transmission            is done but we only measure the interference level on the            xDSL line at the receiver. Finally, the combined DSIN-NEXT            and DSIN-FEXT can be estimated by subtracting the            self-interference level from the level measured at the            receiver.    -   3. an adaptive estimation (training) procedure run when the        modem “turns on”.    -   4. Optimize the spectrum of transmission using the algorithms of        Section 4.5.7 or 4.5.8.    -   5. Transmit and receive data.    -   6. Optional: Periodically update noise and crosstalk estimates        and transmit spectrum from Steps 1-3.

FIG. 27 illustrates a flowchart showing the steps for the optimal andthe suboptimal solution.

4.5.10 Grouping of Bins and Wider Subchannels

The optimal and near-optimal solutions of Sections 4.5.7 and 4.5.8divide the channel into narrow subchannels (bins) and employ theassumptions as discussed in Sections 4.1 and 4.5.3. In the case ofself-interference, the resulting optimal transmit spectrum uses FDS andis “discrete” (a “line spectrum”). Such a transmit spectrum is easilyimplemented via a DMT modulation scheme, but is not easy to implementwith other modulation schemes like PAM, multi-level PAM, or QAM [20]. Inaddition, the DMT scheme can introduce high latency which may be aproblem in some applications. Thus, one may want to use otherlow-latency modulation schemes. In such a scenario, we can combine orgroup FDS bins to form wider subchannels and then employ other broadbandmodulation schemes. This may result in different performance margins butwe believe that the change in margins would not be significant. Analternative broadband modulation scheme like multi-level PAM or QAMwould use a decision feedback equalizer (DFE) [20] at the receiver tocompensate for the channel attenuation characteristic (see Section4.12.4 for further discussion).

FIG. 28 shows one possible way of grouping the bins. The left-hand-sidefigures show the optimal upstream and downstream “discrete” transmitspectra S^(u)(f) and S^(d)(f) as obtained by the algorithm of Section4.5.7. The right-hand-side figures show the same optimal transmitspectra after appropriate grouping of bins resulting in “contiguous”transmit spectra. While grouping, only the bins employing FDS signalingare grouped together and the leftmost bins employing EQPSD signaling areretained as they are. In this particular case, we have grouped the binssuch that the upstream and downstream capacities are equal(C^(u)=C^(d)). The upstream transmit spectrum is completely “contiguous”while the downstream spectrum is “contiguous” except for one “hole” asshown in FIG. 28.

Note: This is not the only way that the bins can be grouped. The binscan be grouped in a variety of different ways giving many differentoptimal transmit spectra. Particular modulation schemes and spectralcompatibility with neighboring services may influence the way bins aregrouped. Further, grouping of bins may lead to different input powersfor opposite directions of transmission.

We look at another possible way of grouping bins such that we achieveequal performance margins and equal upstream and downstream averagepowers. This could be a preferred grouping for symmetric data-rateservices.

Algorithm for “contiguous” optimal transit spectra: Equal margins andequal average powers in both directions:

-   -   1. Solve for the optimal transmit spectrum S^(u)(f) according to        the algorithms in Sections 4.5.7, 4.5.8, or 4.6, where S^(u)(f)        is the water-filling solution (refer to [14] if the spectral        region employs EQPSD or multi-line FDS signaling and to [16] if        the spectral region employs FDS signaling) (see Sections 4.5 and        4.6). This gives a discrete transmit spectrum S^(u)(f).    -   2. Denote the spectral region employing FDS signaling as EFDS        and the spectral region employing EQPSD signaling as E_(EQPSD).    -    Obtain S^(d)(f) from S^(u)(f) by symmetry, i.e.,        S^(d)(f)=S^(u)(f) in EQPSD and multi-line FDS regions and        S^(d)(f)⊥S^(u)(f) in FDS spectral regions. Merge S^(d)(f) and        S^(u)(f) to form S(f) as        S(f)=S ^(u)(f)=S ^(d)(f) ∀f in E _(EQPSD),        S(f)=S ^(u)(f)∪S ^(d)(f) ∀f in E _(FDS),  (29)    -    where ∪ represents the union of the two transmit spectra.    -   3. Estimate bins M_(C)ε(M_(E2F), K], and M_(G)ε(M_(C), K]. Group        the bins of S(f) to obtain upstream and downstream transmit        spectra as $\begin{matrix}        {{S_{opt}^{u}(f)} = \left\{ \begin{matrix}        {S(f)} & {{\forall{f\quad{in}\quad E_{EQPSD}}},{and}} \\        \quad & {{\forall{f\quad{in}\quad{bins}\quad\left( {M_{C},M_{G}} \right\rbrack}},} \\        0 & {{otherwise},}        \end{matrix} \right.} & (30) \\        {{S_{opt}^{d}(f)} = \left( \begin{matrix}        {S(f)} & {{\forall{f\quad{in}\quad E_{EQPSD}}},{and}} \\        \quad & {{\forall{f\quad{in}\quad{bins}\quad\left( {M_{E2F},M_{C}} \right\rbrack}},{and}} \\        \quad & {{\forall{f\quad{in}\quad{bins}\quad\left( {M_{G},K} \right\rbrack}},} \\        0 & {{otherwise}.}        \end{matrix} \right.} & (31)        \end{matrix}$    -   4. Iterate previous step for various choices of M_(C) and M_(G).        The bin M_(C) is chosen such that we get equal performance        margins in both directions of transmission and the bin M_(G) is        chosen such that upstream and downstream directions have equal        average powers.

The resulting transmit spectra S_(opt) ^(e)(f) and S_(opt) ^(d)(f) areanother manifestation of the grouping of bins and yield equalperformance margins (equal capacities) and equal average powers in bothdirections of transmission.

4.5.11 Examples and Results

In this Section, we present some examples and results for the HDSL2service. AGN of −140 dBm/Hz was added to the interference combination inall simulations. Table 1 lists our simulation results performancemargins and compares them with results from [1]. The simulations weredone for the Carrier Serving Area (CSA) loop number 6, which is a 26AWG, 9 kft line with no bridged taps. The column “Our-PAM” refers to ourimplementation using T1E1.4/97-180R1 [11] of the PAM scheme (MONET-PAM)suggested by the authors in [1] using their transmit spectra. We believethe slight differences in margins between MONET-PAM and “Our-PAM” existdue to slight differences in our channel, self-NEXT and self-FEXTmodels. The use of “Our-PAM” margins allows us a fair comparison of ouroptimal results with other proposed transmit spectra. The columns Up andDn refer to the upstream and downstream performance marginsrespectively. The column Optimal refers to the performance marginsobtained using the optimal transmit spectra The column Diff shows thedifference between the performance margins for the optimal transmitspectrum and the MONET-PAM transmit spectrum (using “Our-PAM” margins).A full-duplex bit rate of 1.552 Mbps and a BER of 10⁻⁷ was fixed inorder to get the performance margins. The HDSL2 standards committeedesires a high uncoded margin (preferably more than 6 dB). Table 1 showsthat we achieve very high uncoded margins far exceeding current schemes.

TABLE 1 Uncoded performance margins (in dB) for CSA No. 6: MONET-PAM vs.Optimal. Crosstalk xDSL MONET-PAM “Our-PAM” source service Up Dn Up DnOptimal Diff 49 HDSL HDSL2 9.38 3.14 10.05 3.08 18.75 15.67 39 selfHDSL2 10.3 6.03 11.18 6.00 18.39 12.39 25 T1 HDSL2 19.8 20.3 14.23 20.2921.54 7.31 Bit rate fixed at 1.552 Mbps. Diff = Difference betweenOptimal and worst-case “Our-PAM”.

Table 2 shows the difference between the optimal solution of thesignaling scheme (using the optimal M_(E2F)) and the fast approximatesuboptimal solution (using M_(E2F)=M_(E)) for a variety of interferinglines. The column Diff (in dB) notes the difference in performancemargins between the optimal scheme and the suboptimal scheme. Note thatthere is hardly any difference between the two when self-interferencedominates the total crosstalk. This is a very significant result from animplementation view point for it shows that near-optimal signaling canbe obtained with very little computational effort. The optimal solutionrequires a somewhat complicated optimization over the bins starting fromM_(E) and moving towards the right. Our results clearly indicate thatthe near-optimal solution can give extremely attractive results with nosearch for the optimal bin. Further, this suggests that the optimal binM_(E2F) is closer to M_(E) than M_(F) and so one should search for it tothe immediate right of M_(E).

An optimal upstream transmit spectrum in the case of self-interferenceis illustrated in FIG. 29. The Figure shows the optimal upstreamtransmit spectrum for HDSL2 service in the presence of self-NEXT andself-FEXT from 39 HDSL2 disturbers and AGN of −140 dBm/Hz. Thedownstream transmit spectra for the HDSL2 service are symmetric with theupstream transmit spectra as discussed earlier.

FIG. 30 illustrates optimal “contiguous” transmit spectra for the samecase of 39 self-NEXT and self-FEXT disturbers with AGN of −140 dBm/Hz.The “contiguous” transmit spectra were obtained by grouping the bins asoutlined in Section 4.5.10 (C^(u)=C^(d)). The upstream and downstreamdirections exhibit the same performance margins and use differentpowers.

TABLE 2 Uncoded performance margins (in dB) for CSA No. 6: Optimal vs.Suboptimal. Optimal Fast, xDSL scheme suboptimal Crosstalk sourceservice (dB) M_(E2F) scheme (dB) M_(E) Diff 1 self HDSL2 27.68 11 27.6810 0 10 self HDSL2 21.94 10 21.94 10 0 19 self HDSL2 20.22  8 20.22  8 029 self HDSL2 19.13  8 19.13  8 0 39 self HDSL2 18.39  9 18.39  9 0 10self + 10 HDSL HDSL2 12.11 60 11.46 19 0.65 10 self + 10 T1 HDSL2 7.9227 7.90 23 0.02 Bit rate fixed at 1.552 Mbps. Diff = Difference betweenOptimal and suboptimal scheme.

FIG. 31 illustrates another set of optimal “contiguous” transmit spectrafor the same case of 39 self-NEXT and self-FEXT disturbers with AGN of−140 dBm/Hz. These “contiguous” transmit spectra were obtained bygrouping the bins as outlined in the algorithm of Section 4.5.10 suchthat now we have both equal performance margins (equal capacities) andequal average powers in both directions of transmission.

4.5.12 Spectral Compatibility

When we optimize the capacity of an xDSL service in the presence ofinterferers, we must ensure that the optimized xDSL service is notspectrally incompatible with other services. That is, the performancemargins of other services must not significantly degrade due to thepresence of that xDSL. Our optimal xDSL transmit spectra involvewater-filling (after choosing the appropriate joint signaling strategy).To maximize xDSL capacity we distribute more power in regions of lessinterference and vice versa. This implies the services which interferewith xDSL see less interference in spectral regions where they have morepower and vice versa. This suggests that the spectral compatibilitymargins for other services in the presence of optimized xDSL PSD shouldbe high.

Table 3 lists our simulation results for HDSL2 service and compares themwith results from [1]. The simulations were done for the CSA loop number6 (26 AWG, 9 kft, no bridged taps) and CSA loop number 4 (26 AWG,bridged taps). The column “Our-PAM” refers to our implementation usingT1E1.4/97-180R1 [11] of the PAM scheme (MONET-PAM) suggested by theauthors in [1] using their transmit spectra. We believe the slightdifferences in margins between MONET-PAM and “Our-PAM” exist due to thedifferences in our channel, self-NEXT and self-FEXT models. The columnOptimal lists the performance margins of the xDSL service underconsideration using the optimal transmit spectrum only when HDSL2 is acrosstalk source. The use of “Our-PAM” margins allows us a faircomparison of our optimal margins with the other proposed transmitspectra. From Table 3, we can clearly see that the optimal transmitspectrum has a high degree of spectral compatibility with thesurrounding interfering lines.

Our optimal results in case of self-NEXT and self-FEXT give rise to FDSsignaling, which has a peaky PSD in bins employing FDS. All orthogonalschemes like FDS, TDS, and CDS give self-NEXT rejection and can transmitat the same bit rate. But, using FDS is better than CDS since there is again in the performance margin of the interfering line. We now provethat FDS signaling gives higher spectral compatibility margins thanother orthogonal schemes like CDS.

-   -   Theorem: Let the line under consideration be the signaling line        (with PSD S in a single bin) and the line that interferes with        this line be the interfering line (with PSD s^(u)(f) and        s^(d)(f) in a single bin). Then, using an FDS scheme instead of        CDS scheme for the interfering line results in higher capacity        for the signaling line under an average power constraint and a        Gaussian channel model.        Proof: Consider, as usual the scenario of one single frequency        bin of width W (Hz) as illustrated in FIG. 32. In this Figure, S        is the transmit spectrum of the signaling line under        consideration (for example T1, HDSL, ADSL, etc.), Y and Z        represent the different service interference powers from a        neighboring interfering line (for example HDSL2) and N        represents the lumped channel noise (AGN) and other        different-service interference. There are two cases of interest:

TABLE 3 Spectral-compatibility margins: MONET-PAM vs. Optimal MONET-PAMDn “Our-PAM” Optimal Crosstalk Src XDSL Srvc CSA 6 CSA 4 CSA 6 CSA 4 CSA6 CSA 4 49 HDSL HDSL 8.53 8.09 8.09 7.78 39 HDSL2 Up HDSL 10.1 10.9 9.7410.53 15.44 15.60 39 HDSL2 Dn HDSL 8.28 7.99 7.74 7.53 39 HDSL EC ADSL8.43 9.55 7.84 9.02 39 HDSL2 EC ADSL 9.70 11.7 8.17 10.00 6.93 9.10 49HDSL EC ADSL 8.12 9.24 7.52 8.7 49 HDSL2 HDSL 7.10 6.91 14.95 15.12

-   -   Case 1: The interfering line uses a CDS signaling scheme. In        this case the power in a single bin k (P_(m)) is uniformly        distributed throughout the bin resulting in a flat PSD, i.e.,        s ^(u)(f)=s ^(d)(f)=a.    -    We assume the subchannel frequency responses (1)-(3) and the        notation introduced in (12) and (13). We assume here that the        NEXT and FEXT coupling transfer functions between different        service lines are the same as that for same-service lines. Thus,        we can write the different service interference power in        signaling line bin k as $\quad\begin{matrix}        {\begin{matrix}        {{{{DS}_{N}(f)} + {{DS}_{F}(f)}} = {{{{\overset{\prime}{s}}^{u}(f)}X} + {{s^{d}(f)}F}}} \\        {= {{aX} + {aF}}}        \end{matrix}.} & (32)        \end{matrix}$    -    We define Y and Z as        Y=a(X−F)        Z=2aF.  (33)    -    Using (33) we can write the interference power in (32) as        DS _(N)(f)+DS _(F)(f)=Y+Z.    -   Case 2: The interfering line uses an FDS signaling scheme. In        this case the power in a single bin k (P_(m)) is distributed in        only half the bin, resulting in a peaky PSD, i.e.,        ${s^{u}(f)} = \left\{ {{\begin{matrix}        {{2a},} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\        {0,} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},}        \end{matrix}{and}},{{s^{d}(f)} = \left\{ \begin{matrix}        {0,} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\        {{2a},} & {{{if}\quad\frac{W}{2}} < {f} \leq {W.}}        \end{matrix} \right.}} \right.$    -    We assume the subchannel frequency responses (1)-(3) and the        notation introduced in (12) and (13). We assume here that the        NEXT and FEXT coupling transfer functions between different        service lines are the same as that for same-service lines. Thus,        we can write the different service interference power in        signaling line bin k as $\begin{matrix}        {\quad\begin{matrix}        {{{{DS}_{N}(f)} + {{DS}_{F}(f)}} = {{{s^{u}(f)}X} + {{s^{d}(f)}F}}} \\        {= \left\{ \begin{matrix}        {{2{aX}},} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\        {{2{aF}},} & {{{if}\quad\frac{W}{2}} < {f} \leq {W.}}        \end{matrix} \right.}        \end{matrix}} & (34)        \end{matrix}$    -    Using (33) we can write the interference power in (34) as        ${{{DS}_{N}(f)} + {{DS}_{F}(f)}} = \left\{ \begin{matrix}        {{{2Y} + Z},} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\        {Z,} & {{{if}\quad\frac{W}{2}} < {f} \leq {W.}}        \end{matrix} \right.$

Getting back to the problem, we consider a single signaling line (line1). We divide the signaling line channel into narrow subchannels (orbins) and we analyze a narrow subchannel k. We use the standardassumptions of Section 4.1. We can write the upstream subchannelcapacity of bin k of the signaling line in Case 1 as $\begin{matrix}{{{c_{1}^{u}\left( {{Case}\quad 1} \right)} = {\frac{W}{2\ln\quad 2}\left\{ {{\ln\left\lbrack {1 + \frac{S}{Y + Z + N}} \right\rbrack} + {\ln\left\lbrack {1 + \frac{S}{Y + Z + N}} \right\rbrack}} \right\}}},} & (35)\end{matrix}$and in Case 2 as $\begin{matrix}{{c_{1}^{u}\left( {{Case}\quad 2} \right)} = {\frac{W}{2\ln\quad 2}{\left\{ {{\ln\left\lbrack {1 + \frac{S}{{2Y} + Z + N}} \right\rbrack} + {\ln\left\lbrack {1 + \frac{S}{Z + N}} \right\rbrack}} \right\}.}}} & (36)\end{matrix}$Compute the capacity differences in the two cases as $\begin{matrix}{D = {{{c_{1}^{u}\left( {{Case}\quad 2} \right)} - {c_{1}^{u}\left( {{Case}\quad 1} \right)}} = {\frac{W}{2\ln\quad 2}{{\ln\left\lbrack \frac{\left( {1 + \frac{S}{{2Y} + Z + N}} \right)\left( {1 + \frac{S}{Z + N}} \right)}{\left( {1 + \frac{S}{Y + Z + N}} \right)^{2}} \right\rbrack}.}}}} & (37)\end{matrix}$Taking the partial derivative of D with respect to Y we get$\frac{\partial D}{\partial Y} = {\frac{W}{\ln\quad 2}{{S\left\lbrack {\frac{1}{\left( {Y + Z + N} \right)\left( {Y + Z + N + S} \right)} - \frac{1}{\left( {{2Y} + Z + N} \right)\left( {{2Y} + Z + N + S} \right)}} \right\rbrack}.}}$LetU=Y+Z+N,V=Y+Z+N+S.Note that U, V≧0 and that we can rewrite the partial derivative of Dwith respect to Y as $\begin{matrix}{\frac{\partial D}{\partial Y} = {{\frac{W}{\ln\quad 2}{S\left\lbrack {\frac{1}{UV} - \frac{1}{\left( {U + Y} \right)\left( {V + Y} \right)}} \right\rbrack}} = {{\frac{W}{\ln\quad 2}{S\left\lbrack \frac{Y^{2} + {\left( {U + V} \right)Y}}{{{UV}\left( {U + Y} \right)}\left( {V + Y} \right)} \right\rbrack}} \geq 0.}}} & (38)\end{matrix}$Further, ${\frac{\partial D}{\partial Y}}_{Y = 0} = 0.$The slope of D with respect to Y is always positive and hence, c₁^(u)(Case 2)−c₁ ^(u)(Case 1) is always increasing with Y, which impliesthatc ₁ ^(u)(Case 2)−c ₁ ^(u)(Case 1) ∀Y≧0.When Y<0, i.e., when FEXT is higher than NEXT in a bin (F>X), we canredefine Y and Z asZ=2aX, and, Y=a(F−X).We can then follow the same analysis and show that the capacity c₁^(u)(Case 2) is greater than c₁ ^(u)(Case 1).

Thus, we have proven that FDS scheme rather than CDS scheme forinterfering lines, results in higher capacities for signaling linesunder an average power constraint. Q.E.D.

Interestingly, the power-peaky FDS transmit spectra should be verycompatible with the ADSL standard, since ADSL can balance how many bitsit places in each of its DMT subchannels using a bit loading algorithm[17].

Note: FDS beats CDS in terms of spectral compatibility margins only inthe case of an average power constraint. In the case of a peak powerconstraint in frequency, we may not be able to use a power-peaky schemelike FDS in some spectral regions. Here, we may find that otherorthogonal signaling schemes like CDS offer better spectralcompatibility margins.

4.6 Optimization: Interference from Other Services (DSIN-NEXT andDSIN-FEXT) Plus Self-interference (Self-NEXT and HighSelf-FEXT)—Solution: EQPSD, FDS and Multi-line FDS Signaling

In this scenario we have self-interference (self-NEXT and highself-FEXT) in addition to AGN and DSIN-NEXT and DSIN-FEXT from otherservices (see FIG. 3) in a generic xDSL service. This is the case ofinterest for “GDSL”, “VDSL2”, and HDSL2 (with a small number of lines).

4.6.1 Self-FEXT and Self-NEXT Rejection Using Multi-line FDS

To reject self-FEXT and self-NEXT, we use multi-line FDS (see Section4.3 and FIG. 18). In multi-line FDS we separate each line bytransmitting on each in different frequency bands. This reduces thetransmission bandwidth to 1/M the total channel bandwidth, with M thenumber of lines carrying the service under consideration. Thus,multi-line FDS signaling can increase the capacity only when there are afew number of lines.

We will design a system here that has both self-NEXT and self-FEXTrejection capability. Thus, this serves as the complete solution underthe assumptions in Section 4.1 and the constraints of limited averageinput power (P_(max)) and equal capacity in both directions.

4.6.2 Problem Statement

Maximize the capacity of an xDSL line in the presence of AGN,interference (DSIN-NEXT and DSIN-FEXT) from other services, andself-NEXT and self-FEXT under two constraints:

-   -   1. The average xDSL input power in each direction of        transmission must be limited to P_(max) (Watts), and    -   2. Equal capacity in both directions (upstream and downstream)        for xDSL.        Do this by designing the distribution of energy over frequency        (the transmit spectrum) of the upstream and downstream xDSL        transmissions.

4.6.3 Additional Assumptions

We add the following assumptions to the ones in Section 4.1:

-   -   12. All the M lines carrying the xDSL service are assumed to        have the same channel and noise characteristics and face the        same interference combination in both transmission directions        (upstream and downstream). Refer to Section 4.7 for results when        this does not hold true.    -   13. The coupling transfer functions of NEXT and FEXT        interference are symmetrical between neighboring services. For        example, each line has the same self-NEXT transfer function        H_(N)(f) and self-FEXT transfer function H_(F)(f) for computing        coupling of interference power with any other line. However, we        develop some results in Section 4.7 when there are different        NEXT and FEXT coupling transfer functions between lines.

4.6.4 Signaling Scheme

The level of self-NEXT and self-FEXT varies over frequency (recall FIG.17). In regions of low self-NEXT and low self-FEXT, EQPSD signaling isthe best choice. In spectral regions of high self-NEXT but lowself-FEXT, orthogonal signaling scheme like FDS is preferred (due to itsself-NEXT rejection, as we saw in Section 4.5). But, in regions of highself-FEXT, multi-line FDS signaling might be required for gainingcapacity.

Key to our scheme is that the upstream and downstream transmissions ofeach of the M lines use different transmit spectra.

4.6.5 Solution Using EQPSD and FDS Signaling: All Frequency Bins

First, we assume that self-FEXT is small and then, using EQPSD or FDSsignaling in each bin, we find the solution for all frequency bins asoutlined in Sections 4.5.4-4.5.8. Thus, we obtain the optimal (orsuboptimal) EQPSD to FDS switch-over bin M_(E2F) under the low self-FEXTassumption.

Next, we relax the self-FEXT assumption and open the possibility ofmulti-line FDS. We search each bin to see if we need to switch fromEQPSD to multi-line FDS or FDS to multi-line FDS. This may notnecessarily yield the optimal solution for the transmit spectrum giventhat we use a joint signaling scheme comprising of the three signalingschemes (EQPSD, FDS and multi-line FDS). But, this analysis is tractableand gives significant gains in channel capacity and is presented next.

4.6.6 Switch to Multi-line FDS: One Frequency Bin

Consider the case of M lines with significant self-FEXT interferencebetween them. We divide the channel into several equal bandwidth (W Hz)bins (see FIG. 16) and perform our analysis on one frequency bin kassuming subchannel frequency responses (1)-(3). We employ the notationintroduced in (12) and (13). Let s₁ ^(u)(f) denote the PSD in bin k ofline 1 upstream direction and s₁ ^(d)(f) denote the PSD in bin k of line1 downstream direction (recall the notation introduced in Section 4.1,Item 9). Let P_(m) be the average power over the frequency range [0, W].

Next, we determine when we need to switch to multi-line FDS in a givenbin to completely reject self-FEXT:

EQPSD to multi-line FDS: FIG. 33 illustrates the two possible signalingschemes EQPSD and multi-line FDS in bin k of each line for the case ofM=3 lines. We will consider line 1 for our capacity calculations. Line 1upstream and downstream capacities for EQPSD signaling are denoted byc_(1,EQPSD) ^(u) and c_(1,EQPSD) ^(d) respectively. Similarly, line 1upstream and downstream capacities for multi-line FDS signaling aredenoted by c_(1,MFDS) ^(u) and c_(1,MFDS) ^(d) respectively. Since theupstream and downstream transmit spectra of line 1 in bin k for EQPSDand multi-line FDS are the same, we have:

-   -   c_(1,EQPSD) ^(u)=c_(1,EQPSD) ^(d), c_(1,MFDS) ^(u)=c_(1,MFDS)        ^(d)        Thus, we will consider only the upstream capacities in our        future discussion.

Under the Gaussian channel assumption, we can define the EQPSD upstreamcapacity (in bps) as $\begin{matrix}{{{c_{1,{EQPSD}}^{u} = {W\quad{\log_{2}\left\lbrack {1 + \frac{{s_{1}^{u}(f)}H}{N + {s_{1}^{d}X} + {s_{1}^{u}F}}} \right\rbrack}}},{where}}{{s_{1}^{u}(f)} = {{s_{1}^{d}(f)} = \left\{ \begin{matrix}{\frac{P_{m}}{W},} & {{{{if}\quad{f}} \in \left\lbrack {0,W} \right\rbrack},} \\0 & {{otherwise}.}\end{matrix} \right.}}} & (39)\end{matrix}$

Let $G = \frac{2P_{m}}{WN}$denote the SNR in the bin. Then we can rewrite c_(1,EQPSD) ^(u) as$\begin{matrix}{c_{1,{EQPSD}}^{u} = {W\quad{{\log_{2}\left\lbrack {1 + \frac{GH}{2 + {GX} + {GF}}} \right\rbrack}.}}} & (40)\end{matrix}$

Similarly, we can define the multi-line FDS upstream capacity (in bps)as $\begin{matrix}{{{c_{1,{MFDS}}^{u} = {\frac{W}{M}\quad{\log_{2}\left\lbrack {1 + \frac{{s_{1}^{u}(f)}H}{N}} \right\rbrack}}},{where}}{{s_{1}^{u}(f)} = \left\{ \begin{matrix}{\frac{{MP}_{m}}{W},} & {{{{if}\quad{f}} \in \left\lbrack {0,\frac{W}{M}} \right\rbrack},} \\{0,} & {{otherwise},}\end{matrix} \right.}} & (41)\end{matrix}$and $G = \frac{2P_{m}}{WN}$is the SNR in the bin. Then we can rewrite c_(1,MFDS) ^(u) as$\begin{matrix}{{c_{1,{MFDS}}^{u} = {\frac{W}{M}{\log_{2}\left\lbrack {1 + {\frac{M}{2}{GH}}} \right\rbrack}}},} & (42)\end{matrix}$

Define the difference between the two capacities asD=c _(1,MFDS) ^(u) −c _(1,EQPSD) ^(u).  (43)

We wish to determine when it is better to do multi-line FDS than EQPSD,i.e., when is the capacity c_(1,MFDS) ^(u) greater than c_(1,EQPSD)^(u). This means we need a condition for when D>0. Substituting from(40) and (42) into (43) we get D>0 iff $\begin{matrix}{F > {\frac{\left\lbrack {2 + {G\left( {X + H} \right)}} \right\rbrack - {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{1}{M}}\left( {2 + {GX}} \right)}}{G\left( {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{1}{M}} - 1} \right)}.}} & (44)\end{matrix}$

Similarly, EQPSD is better (gives higher capacity) than multi-line FDSwhen D<0, i.e., iff $\begin{matrix}{F < {\frac{\left\lbrack {2 + {G\left( {X + H} \right)}} \right\rbrack - {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{1}{M}}\left( {2 + {GX}} \right)}}{G\left( {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{1}{M}} - 1} \right)}.}} & (45)\end{matrix}$

We can combine (44) and (45) into one test condition that tells us thesignaling scheme to use in a single frequency bin $\begin{matrix}{F\begin{matrix}\begin{matrix}\begin{matrix}{{multi}\text{-}{line}\quad{FDS}} \\ > \end{matrix} \\ < \end{matrix} \\{EQPSD}\end{matrix}{\frac{\left\lbrack {2 + {G\left( {X + H} \right)}} \right\rbrack - {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{1}{M}}\left( {2 + {GX}} \right)}}{G\left( {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{1}{M}} - 1} \right)}.}} & (46)\end{matrix}$

FDS to multi-line FDS: FIG. 34 illustrates the two possible signalingschemes FDS and multi-line FDS in bin k of each line for the case of M=3lines. We will consider line 1 for our capacity calculations. Line 1upstream and downstream capacities for FDS signaling are denoted byc_(1,FDS) ^(u) and c_(1,FDS) ^(d) respectively. Similarly, line 1upstream and downstream capacities for multi-line FDS signaling aredenoted by c_(1,MFDS) ^(u) and c_(1,MFDS) ^(d) respectively. Since theupstream and downstream transmit spectra of line 1 in bin k for EQPSDand multi-line FDS are the same, we have:

-   -   c_(1,FDS) ^(u)=c_(1,FDS) ^(d), c_(1,MFDS) ^(u)=c_(1,MFDS) ^(d)        Thus, we will consider only the upstream capacities in our        future discussion. Under the Gaussian channel assumption we can        define the FDS upstream capacity (in bps) as $\begin{matrix}        {{c_{1,{FDS}}^{u} = {\frac{W}{2}{\log_{2}\left\lbrack {1 + \frac{{s_{1}^{u}(f)}H}{N + {s_{1}^{u}F}}} \right\rbrack}}},} & (47) \\        {where} & \quad \\        {{s_{1}^{u}(f)} = \left\{ \begin{matrix}        {\frac{2P_{m}}{W},} & {{{{if}\quad{f}} \in \left\lbrack {0,\frac{W}{2}} \right\rbrack},} \\        {0,} & {{otherwise}.}        \end{matrix} \right.} & \quad        \end{matrix}$        Let $G = \frac{2\quad{Pm}}{WN}$        denote the SNR in the bin. Then we can rite c_(1,FDS) ^(u) as        $\begin{matrix}        {c_{1,{FDS}}^{u} = {\frac{W}{2}{{\log_{2}\left\lbrack {1 + \frac{GH}{1 + {GF}}} \right\rbrack}.}}} & (48)        \end{matrix}$        Similarly, we can define the multi-line FDS upstream capacity        (in bps) as $\begin{matrix}        \begin{matrix}        {{c_{1,{MFDS}}^{u} = {\frac{W}{M}{\log_{2}\left\lbrack {1 + \frac{{s_{1}^{u}(f)}H}{N}} \right\rbrack}}},} \\        {where} \\        {{s_{1}^{u}(f)} = \left\{ \begin{matrix}        {\frac{{MP}_{m}}{W},} & {{{{if}\quad{f}} \in \left\lbrack {0,\frac{W}{M}} \right\rbrack},} \\        {0,} & {{otherwise},}        \end{matrix} \right.}        \end{matrix} & (49)        \end{matrix}$        and $G = \frac{2P_{m}}{WN}$        is the SNR in the bin. Then we can rewrite c_(1,MFDS) ^(u) as        $\begin{matrix}        {{c_{1,{MFDS}}^{u} = {\frac{W}{M}{\log_{2}\left\lbrack {1 + {\frac{M}{2}{GH}}} \right\rbrack}}},} & (50)        \end{matrix}$        Define the difference between the two capacities as        D=c _(1,MFDS) ^(u) −c _(1,FDS) ^(u)  (51)        We wish to find out when it is more appropriate to perform        multi-line FDS than FDS, i.e., when the capacity c_(1,MFDS) ^(u)        is greater than c_(1,FDS) ^(u). For this, we need a condition        for when D>0. Substituting from (48) and (50) into (51) we get        D>0 iff $\begin{matrix}        {F > {\frac{\left( {1 + {GH}} \right) - \left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{2}{M}}}{G\left( {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{2}{M}} - 1} \right)}.}} & (52)        \end{matrix}$        Similarly, FDS is better (gives higher capacity) than multi-line        FDS when D<0, i.e., iff $\begin{matrix}        {F < {\frac{\left( {1 + {GH}} \right) - \left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{2}{M}}}{G\left( {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{2}{M}} - 1} \right)}.}} & (53)        \end{matrix}$        We can combine (52) and (53) into one test condition which tells        us the signaling scheme to use $\begin{matrix}        {F\begin{matrix}        {{multi}\text{-}{line}\quad{FDS}} \\         > \\         < \\        {FDS}        \end{matrix}{\frac{\left( {1 + {GH}} \right) - \left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{2}{M}}}{{G\left( {\left( {1 + {\frac{M}{2}{GH}}} \right)^{\frac{2}{M}} - 1} \right)}.}.}} & (54)        \end{matrix}$

Thus, we can write the generic upstream capacity c₁ ^(u) for bin k ofline 1 as $\begin{matrix}{c_{1}^{u} = \left\{ \begin{matrix}{{W\quad{\log_{2}\left\lbrack {1 + \frac{P_{m}H}{{NW} + {P_{m}\left( {X + F} \right)}}} \right\rbrack}},} & {{{if}\quad{EQPSD}},} \\{{\frac{W}{2}{\log_{2}\left\lbrack {1 + \frac{P_{m}H}{{N\frac{W}{2}} + {P_{m}F}}} \right\rbrack}},} & {{{if}\quad{FDS}},} \\{{\frac{W}{M}{\log_{2}\left\lbrack {1 + \frac{{MP}_{m}H}{WN}} \right\rbrack}},} & {{if}\quad{multi}\text{-}{line}\quad{{FDS}.}}\end{matrix} \right.} & (55)\end{matrix}$

4.6.7 Switch to Multi-line FDS: All Frequency Bins

We saw in the previous Section how to determine if we need to switch tomulti-line FDS from EQPSD or FDS in a given bin. We already have theoptimal solution assuming EQPSD and FDS signaling scheme (from Section4.5). Now, we apply the conditions (46) and (54) to each bin k.Interestingly, due to the assumed monotonicity of self-FEXT, self-NEXTand channel transfer function, we can divide the frequency axis (all Kbins) into 4 major regions:

-   -   1. Using test condition (46), we find that bins [1, M_(E2MFDS)]        employ EQPSD signaling.    -   2. Using test condition (46), we find that bins [M_(E2MFDS)+1,        M_(MFDS2FDS)] employ multi-line FDS signaling. Note that        M_(MFDS2FDS)=M_(E2F) obtained from optimization procedure of        Section 4.6.5.    -   3. Using test condition (54), we find that bins [M_(MFDS2FDS)+1,        M_(FDS2MFDS)] employ FDS signaling.    -   4. Using test condition (54), we find that bins [M_(FDS2MFDS)+1,        K] employ multi-line FDS signaling.

FIG. 35 illustrates the 3 bins M_(E2MFDS), M_(MFDS2FDS) and M_(FDS2MFDS)and the EQPSD, FDS and multi-line FDS regions. In practice we mainly see2 scenarios:

-   -   1. If M_(E2MFDS)<M_(MFDS2FDS) then M_(FDS2MFDS)=M_(MFDS2FDS),        and we get only 2 distinct spectral regions as shown in FIG. 36:        -   (a) Bins [1, M_(E2MFDS)] employ EQPSD signaling.        -   (b) Bins [M_(E2MFDS)+1, K] employ multi-line FDS signaling.    -    FDS signaling is not employed in this case.    -   2. If M_(E2MFDS)=M_(MFDS2FDS)=M_(E2F) then we get 3 distinct        spectral regions as shown in FIG. 37:        -   (a) Bins [1, M_(MFDS2FDS)] employ EQPSD signaling.        -   (b) Bins [M_(MFDS2FDS)+1, M_(FDS2MFDS)] employ FDS            signaling.        -   (c) Bins [M_(FDS2MFDS)+1, K] employ multi-line FDS            signaling.    -    There is no switch to multi-line FDS signaling within the EQPSD        signaling region (bins [1, M_(E2F)]).        Note that the bin M_(MFDS2FDS)=M_(E2F) is fixed from the        optimization procedure from Section 4.6.5.

4.6.8 Special Case: Performance of 2 Lines

Often in practice we may have only two twisted pair lines carrying thesame service and interfering with each other. It is important to derivethe optimal transmit spectrum for such a scenario. In this Section wefocus on this special case of only 2 lines. We will see that in thiscase it is optimal to perform either multi-line FDS or EQPSD signalingin each bin. In this scenario with arbitrary self-FEXT and self-NEXT weeasily see that there is no need to perform FDS signaling (rejectself-NEXT only) as multi-line FDS rejects both self-NEXT and self-FEXTwhile achieving the same capacity as FDS. Thus, we choose between EQPSDand multi-line FDS signaling schemes for each bin to achieve the optimaltransmit spectrum.

Let S₁ ^(u)(f) and S₁ ^(d)(f) denote the upstream and downstreamtransmit spectra of line 1 and S₂ ^(u)(f) and S₂ ^(d)(f) denote theupstream and downstream transmit spectra of line 2 respectively. Let theline 1 upstream capacity be C₁ ^(u) and let the line 2 downstreamcapacity be C₂ ^(d). Under the Gaussian channel assumption, we can writethese capacities (in bps) as $\begin{matrix}{{C_{1}^{u} = {\sup\limits_{{S_{1}^{u}{(f)}},{S_{2}^{d}{(f)}},{S_{2}^{u}{(f)}}}{\int_{0}^{\infty}{{\log_{2}\left\lbrack {1 + \frac{{{H_{C}(f)}}^{2}{S_{1}^{u}(f)}}{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)} + {{{H_{N}(f)}}^{2}{S_{2}^{d}(f)}} + {{{H_{F}(f)}}^{2}{S_{2}^{u}(f)}}}} \right\rbrack}{\mathbb{d}f}}}}},{and}} & (56) \\{C_{2}^{d} = {\sup\limits_{{S_{2}^{d}{(f)}},{S_{1}^{u}{(f)}},{S_{1}^{d}{(f)}}}{\int_{0}^{\infty}{{\log_{2}\left\lbrack {1 + \frac{{{H_{C}(f)}}^{2}{S_{2}^{d}(f)}}{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)} + {{{H_{N}(f)}}^{2}{S_{1}^{u}(f)}} + {{{H_{F}(f)}}^{2}{S_{1}^{d}(f)}}}} \right\rbrack}{{\mathbb{d}f}.}}}}} & (57)\end{matrix}$The supremum is taken over all possible S₁ ^(u)(f), S₂ ^(u)(f), S₁^(d)(f) and S₂ ^(d)(f) satisfyingS ₁ ^(u)(f)≧0, S ₁ ^(d)(f)≧0, S ₂ ^(u)(f)≧0, S ₂ ^(d)(f)≧0, ∀f,and the average power constraints for the two directions $\begin{matrix}{{{2{\int_{0}^{\infty}{{S_{1}^{u}(f)}{\mathbb{d}f}}}} \leq P_{m\quad{ax}}},{{{and}\quad 2{\int_{2}^{\infty}{{S_{2}^{d}(f)}{\mathbb{d}f}}}} \leq {P_{m\quad{ax}}.}}} & (58)\end{matrix}$

We employ multi-line FDS (S₁ ^(u)(f) and S₁ ^(d)(f) orthogonal to S₂^(u)(f) and S₂ ^(d)(f)) in spectral regions where the self-FEXT is largeenough and EQPSD in the remaining spectrum. This gives optimalperformance.

To ease our analysis, as usual, we divide the channel into several equalbandwidth subchannels (bins) (see FIG. 16) and continue our design andanalysis on one frequency bin k assuming subchannel frequency responses(1)-(3). We use notation introduced in (12) and (13). Let s₁ ^(u)(f) ands₁ ^(d)(f) denote the PSDs in bin k of line 1 upstream and downstreamdirections and s₂ ^(u)(f) and s₂ ^(d)(f) denote the PSDs in bin k ofline 2 upstream and downstream directions. The corresponding capacitiesof the subchannel k are denoted by c₁ ^(u), c₁ ^(d), c₂ ^(u) and c₂^(d).

We desire a signaling scheme that can have multi-line FDS, EQPSD and allcombinations in between in each frequency bin. Therefore we divide eachbin in half⁵ and define the upstream and downstream transmit spectra asfollows (see FIG. 38): $\begin{matrix}{{s_{1}^{u}(f)} = {{s_{1}^{d}(f)} = \left\{ \begin{matrix}{\alpha\frac{2P_{m}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\left( {1 - \alpha} \right)\frac{2P_{m}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix} \right.}} & (59) \\{and} & \quad \\{{s_{2}^{u}(f)} = {{s_{2}^{d}(f)} = \left\{ \begin{matrix}{\left( {1 - \alpha} \right)\frac{2P_{m}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\alpha\frac{2P_{m}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix} \right.}} & (60)\end{matrix}$Here, P_(m) is the average power over frequency range [0, W] in bin kand 0.5≦α≦1. In this discussion we will only use the PSDs s₁ ^(u)(f) ands₂ ^(d)(f). When α=0.5, s₁ ^(u)(f)=₂ ^(d)(f) ∀fε[0, W] (EQPSDsignaling); when α=1, s₁ ^(u)(f) and s₂ ^(d)(f) are disjoint (multi-lineFDS signaling). The PSDs s₁ ^(u)(f) and s₂ ^(d)(f) are “symmetrical” orpower complementary to each other. This ensures the capacities of thetwo lines are equal (c₁ ^(u)=c₂ ^(d)). The factor α controls the powerdistribution in the bin and W is the bandwidth of the bin.

⁵The power split-up in a bin does not necessarily have to be 50% to theleft side of the bin and 50% to the right side of the bin as shown inFIG. 38. In general any 50%—50% power complementary split-up betweendifferent-line bins will work.

Next, we show that the optimal signaling strategy uses only multi-lineFDS or EQPSD in each subchannel.

The achievable rate for one frequency bin can be written as$\begin{matrix}{{{R_{A}\left( {{s_{1}^{u}(f)},{s_{2}^{d}(f)},{s_{2}^{u}(f)}} \right)} = {\int_{0}^{W}{{\log_{2}\left\lbrack {1 + \frac{{s_{1}^{u}(f)}H}{N + {{s_{2}^{d}(f)}X} + {{s_{2}^{u}(f)}F}}} \right\rbrack}{\mathbb{d}f}}}},{then}} & (61) \\{{c_{1}^{u} = {\max\limits_{0.5 \leq \alpha \leq 1}{{R_{A}\left( {{s_{1}^{u}(f)},{s_{2}^{d}(f)},{s_{2}^{u}(f)}} \right)}\quad{and}}}}{c_{2}^{d} = {\max\limits_{0.5 \leq \alpha \leq 1}{{R_{A}\left( {{s_{2}^{d}(f)},{s_{1}^{u}(f)},{s_{2}^{u}(f)}} \right)}.}}}} & (62)\end{matrix}$Due to the power complementarity of s₁ ^(u)(f) and s₂ ^(d)(f), thechannel capacities are equal (c₁ ^(u)=c₂ ^(d)). Therefore, we will onlyconsider the upstream capacity c₁ ^(u) expression. Further, we will useR_(A) for R_(A)(s₁ ^(u)(f), s₂ ^(d)(f), s₂ ^(u)(f)) in the remainder ofthis Section. Substituting for the PSDs from (59) and (60) into (61) andusing (62) we get the following expression for the upstream capacity$\begin{matrix}{c_{1}^{u} = {\frac{W}{2}{\max\limits_{0.5 \leq \alpha \leq 1}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\frac{\alpha\quad 2P_{m}H}{W}}{N + \frac{\left( {1 - \alpha} \right)2\quad P_{m}X}{W} + \frac{\left( {1 - \alpha} \right)2P_{m}F}{W}}} \right\rbrack} + {\log_{2}\left\lbrack \quad{1 + \frac{\frac{\left( {1 - \alpha} \right)\quad 2P_{m}H}{W}}{N + \frac{\alpha\quad 2P_{m}X}{W} + \frac{\alpha\quad 2P_{m}F}{W}}} \right\rbrack}} \right\}.}}}} & (63)\end{matrix}$Let $G = \frac{2P_{m}}{WN}$denote the SNR in the bin. Then, we can rewrite (63) as $\begin{matrix}{c_{1}^{u} = {\frac{W}{2}{\max\limits_{0.5 \leq \alpha \leq 1}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\alpha\quad{GH}}{1 + {\left( {1 - \alpha} \right){GX}} + {\left( {1 - \alpha} \right){GF}}}} \right\rbrack} + {\log_{2}\left\lbrack {1 + \frac{\left( {1 - \alpha} \right)\quad{GH}}{1 + {\alpha\quad{GX}} + {\alpha\quad{GF}}}} \right\rbrack}} \right\}.}}}} & (64)\end{matrix}$

Using (62) and differentiating the achievable rate (R_(A)) expression in(64) with respect to α gives us $\begin{matrix}{{\frac{\partial R_{A}}{\partial\alpha} = {{\left( {{2\alpha} - 1} \right)\left\lbrack {{2\left( {X + F} \right)^{2}} - H} \right\rbrack}L}},} & (65)\end{matrix}$with L>0 ∀αε(0,1]. Setting the derivative to zero gives us the singlestationary point α=0.5. Thus, the achievable rate R_(A) is monotonic inthe interval αε(0.5, 1] (see FIG. 24). If the value α=0.5 corresponds toa maximum of R_(A), then it is optimal to perform EQPSD signaling inthis bin. If the value α=0.5 corresponds to a minimum of R_(A), then themaximum of R_(A) is achieved by the value α=1, meaning it is optimal toperform multi-line FDS signaling in this bin. No other values of α arean optimal option (see FIG. 39).

The quantity α=0.5 corresponds to a maximum of R_(A) (EQPSD) if and onlyif $\frac{\partial R_{A}}{\partial\alpha} < 0$∀αε(0.5, 1]. For all αε(0.5, 1], the quantity (2α−1) is positive and$\frac{\partial R_{A}}{\partial\alpha}$is negative iff (see (65))2(X+F)+G(X+F)² −H<0.This implies that $\begin{matrix}{G < {\frac{H - {2\left( {X + F} \right)}}{\left( {X + F} \right)^{2}}.}} & (66)\end{matrix}$

In a similar fashion α=0.5 corresponds to a minimum of R_(A) if and onlyif $\frac{\partial R_{A}}{\partial\alpha} > 0$∀fαε(0.5, 1]. This implies that α=1 corresponds to α=(multi-line FDS)since there is only one stationary point in the interval αε[0.5, 1] (seeFIG. 24). For all αε(0.5, 1], $\frac{\partial R_{A}}{\partial\alpha}$is positive iff2(X+F)+G(X+F)² −H>0.This implies that $\begin{matrix}{G > {\frac{H - {2\left( {X + F} \right)}}{\left( {X + F} \right)^{2}}.}} & (67)\end{matrix}$

The above statements can be summed in a test condition to determine thesignaling nature (multi-line FDS or EQPSD) in a given bin. Using (66)and (67) we can write $\begin{matrix}{G = {\frac{2P_{m}}{NW}\begin{matrix}\begin{matrix}\begin{matrix}{{multi}\text{-}{line}\quad{FDS}} \\ > \end{matrix} \\ < \end{matrix} \\{EQPSD}\end{matrix}{\frac{H - {2\left( {X + F} \right)}}{\left( {X + F} \right)^{2}}.}}} & (68)\end{matrix}$

Thus, we can write the upstream capacity c₁ ^(u) in a frequency bin k as$\begin{matrix}{c_{1}^{u} = \left\{ \begin{matrix}{{W\quad{\log_{2}\left\lbrack {1 + \frac{P_{m}H}{{NW} + {P_{m}\left( {X + F} \right)}}} \right\rbrack}},} & {{{{if}\quad\alpha} = 0.5},} \\{{\frac{W}{2}{\log_{2}\left\lbrack {1 + \frac{2P_{m}H}{NW}} \right\rbrack}},} & {{{if}\quad\alpha} = 1.}\end{matrix} \right.} & (69)\end{matrix}$

Note: It is globally optimal to employ either multi-line FDS or EQPSDsignaling; that is, α=0.5 or 1, only in the case of 2 lines.

4.6.9 Flow of the Scheme

-   -   1. Perform steps 1-3 of Section 4.5.9.    -   2. Compute bins M_(E2MFDS), M_(MFDS2FDS) and M_(FDS2MFDS) and        employ signaling schemes in bins as described in Section 4.6.5.    -   3. Transmit and receive data.    -   4. Optional: Periodically update noise and crosstalk estimates        and transmit spectrum from Steps 1-3 of Section 4.5.9. Repeat        Step 2 from above.

FIG. 40 gives a flowchart to obtain the optimal transmit spectrum usingEQPSD, FDS, and multi-line FDS (MFDS) signaling in the presence ofself-interference (self-NEXT and self-FEXT), DSIN-NEXT, DSIN-FEXT andAGN.

4.6.10 Examples and Results

Optimal transmit spectra were used in all examples to computeperformance margins and channel capacities.

HDSL2 service: Table 4 lists our simulation results performance marginsand channel capacities using the EQPSD, FDS and multi-line FDS signalingschemes.

Notes:

-   -   1. Sampling frequency f_(s)=1000 kHz, Bin width W=2 kHz and        number of subchannels K=250. Average input power of 20 dBm in        each transmission direction.    -   2. C_(i) ^(u) denotes the upstream capacity of line i using        EQPSD and FDS signaling only and C₁ ^(u)(MFDS) denotes the        upstream capacity of line i using EQPSD, FDS and multi-line FDS        signaling schemes. All the rates are in Mbps.    -   3. The column Margin lists the performance margin when the bit        rate is fixed at 1.552 Mbps. In each row in the top half the        capacity is fixed at C_(i) ^(u)=1.5520 and in the bottom half        the capacity is fixed at C_(i) ^(u)(MFDS)=1.5520.

TABLE 4 Uncoded performance margins (in dB) and channel capacities (inMbps) using EQPSD, FDS and multi-line FDS for HDSL2 (CSA No. 6). XtalkSrc M_(E2MFDS) M_(MFDS2FDS) M_(FDS2MFDS) C_(i) ^(u) C_(i) ^(u) (MFDS)Margin Diff 1 HDSL2 8 11  11 1.5520 2.3763 27.682 9.852 1 HDSL2 0 0  00.8027 1.5520 37.534 2 HDSL2 9 9 30 1.5520 1.8293 25.934 4.543 2 HDSL2 44 19 1.1861 1.5520 30.477 3 HDSL2 8 8 112  1.5520 1.6067 24.910 0.985 3HDSL2 7 7 100  1.4792 1.5520 25.791 4 HDSL2 8 8 246  1.5520 1.552024.186 0 Diff = Difference between bottom half and top half of each rowof Margin.

-   -   4. The column Diff denotes the gain in performance margins        between using EQPSD and FDS versus EQPSD, FDS and multi-line FDS        signaling, i.e., the difference in margins between the bottom        half and top half of each row.    -   5. Each HDSL2 line contributes NEXT and FEXT calculated using        2-piece Unger model [8].    -   6. These runs were done with no different service (DS)        interferers. The results would vary depending on the particular        DS interferer(s) present.

Conclusions:

-   -   1. Significant gains in margin for small number of lines. The        gains decrease with increase in number of lines.    -   2. There is no gain in margin using multi-line FDS for 5 or more        lines (4 Crosstalk disturbers) for these line and interference        models.

“GDSL” service: Table 5 lists our simulation results performance marginsand channel capacities using the EQPSD, FDS and multi-line FDS signalingschemes in the case of “GDSL”.

TABLE 5 Uncoded performance margins (in dB) and channel capacities (inMbps) using EQPSD, FDS and multi-line FDS for “GDSL” (3 kft line). XtalkSrc M_(E2MFDS) M_(MFDS2FDS) M_(FDS2MFDS) C_(i) ^(u) C_(i) ^(u) (MFDS)Margin Diff 1 GDSL 505 1253 1253 25.0046 31.6188 8.21 8.49 1 GDSL 245 981  981 16.5141 25.0007 16.70 2 GDSL 952 1214 1214 25.0007 27.39236.13 2.91 2 GDSL 825 1116 1116 22.0076 25.0030 9.04 3 GDSL 1186  12121212 25.0004 25.6686 5.05 0.75 3 GDSL 1145  1186 1186 24.2172 25.00085.80 4 GDSL 1222  1222 2000 25.0018 25.0018 4.37 0 Diff = Differencebetween bottom half and top half of each row of Margin.

Notes:

-   -   1. Sampling frequency f_(s)=8000 kHz, Bin width W=2 kHz and        number of subchannels K=2000. Average input power of 20 dBm in        each transmission direction.    -   2. C_(i) ^(u) denotes the upstream capacity of line i using        EQPSD and FDS signaling only and C_(i) ^(u)(MFDS) denotes the        upstream capacity of line i using EQPSD, FDS and multi-line FDS        signaling schemes. All the rates are in Mbps.    -   3. The column Margin lists the performance margin when the bit        rate is fixed at 25 Mbps. In each row in the top half the        capacity is fixed at C_(i) ^(u)=25 and in the bottom half the        capacity is fixed at C_(i) ^(u)(MFDS)=25.    -   4. The column Diff denotes the gain in performance margins        between using EQPSD and FDS versus EQPSD, FDS and multi-line FDS        signaling, i.e., the difference in margins between the bottom        half and top half of each row.    -   5. Each “GDSL” line contributes self-NEXT and self-FEXT        calculated using 2-piece Unger model [8]. In “GDSL” case the        self-FEXT level is more dominant than self-NEXT. To model this        we take only 1% of the self-NEXT power calculated using 2-piece        Unger model in our simulations.    -   6. These runs were done with no different service (DS)        interferers. The results would vary depending on the particular        DS interferer(s) present.

Conclusions:

-   -   1. Significant gains in margin for small number of lines. The        gains decrease with increase in number of lines.    -   2. There is no gain in margin using multi-Line FDS for 5 or more        lines (4 Crosstalk disturbers) for these line and interference        models.

“VDSL2” service: Table 6 lists our simulation results performancemargins and channel capacities using the EQPSD, FDS and multi-line FDSsignaling schemes in the case of “VDSL2”.

Notes:

-   -   1. Sampling frequency f_(s)=8000 kHz, Bin width W=2 kHz and        number of subchannels K=2000. Average input power of 20 dBm in        each transmission direction.    -   2. C_(i) ^(u) denotes the upstream capacity of line i using        EQPSD and FDS signaling only and C_(i) ^(u)(° FDS) denotes the        upstream capacity of line i using EQPSD, FDS and multi-line FDS        signaling schemes. All the rates are in Mbps.    -   3. The column Margin lists the performance margin when the bit        rate is fixed at 12.4 Mbps. In each row in the top half the        capacity is fixed at C_(i) ^(u)=12.4 and in the bottom half the        capacity is fixed at C_(i) ^(u)(MFDS)=12.4.    -   4. The column Diff denotes the gain in performance margins        between using EQPSD and FDS versus EQPSD, FDS and multi-line FDS        signaling, i.e., the difference in margins between the bottom        half and top half of each row.    -   5. Each VDSL2 line contributes self-NEXT and self-FEXT        calculated using 2-piece Unger model [8]. In VDSL2 case        self-NEXT and self-FEXT both are high but self-NEXT dominates        self-FEXT.

TABLE 6 Uncoded performance margins (in dB) and channel capacities (inMbps) using EQPSD, FDS and multi-line FDS for “VDSL2” (3 kft line).Xtalk Src M_(E2MFDS) M_(MFDS2FDS) M_(FDS2MFDS) C_(i) ^(u) C_(i) ^(u)(MFDS) Margin Diff 1 VDSL2  58 236 236 12.4011 24.8234 16.022 18.913 1VDSL2  8  50  50  2.5552 12.4001 34.935 2 VDSL2 160 219 219 12.400318.8073 14.074 13.476 2 VDSL2  46  78  78  4.4478 12.4036 27.550 3 VDSL2217 217 217 12.4028 15.6002 12.985 7.765 3 VDSL2 127 127 127  7.336512.4002 20.750 4 VDSL2 219 219 553 12.4016 13.7787 12.250 3.275 4 VDSL2179 179 359 10.1474 12.4012 15.525 5 VDSL2 224 224 1014  12.4014 12.903911.705 1.005 5 VDSL2 211 211 878 11.6945 12.4014 12.710 6 VDSL2 231 2311455  12.4025 12.5278 11.280 0.212 6 VDSL2 229 229 1412  12.2521 12.401811.492 7 VDSL2 240 240 1880  12.4004 12.4049 10.945 0.007 7 VDSL2 240240 1878  12.3954 12.4001 10.952 Diff = Difference between bottom halfand top half of each row of Margin.

Conclusions:

-   -   1. Significant gains in margin for small number of lines. The        gains decrease with increase in number of lines.    -   2. There is no gain in margin using multi-line FDS for 9 or more        lines (8 crosstalk disturbers). These runs were done with no        different service (DS) interferers. The results would vary        depending on the particular DS interferer present.        4.7 Joint Signaling for Lines Differing in Channel, Noise and        Interference Characteristics

We have so far looked at a scenario where all the lines in a binder havethe same channel characteristics and experience similar noise andinterference characteristics in both directions of transmission. Theseassumptions made the signaling scheme solutions more tractable. We alsoneed to look at a scenario between neighboring lines in binder groupswhere the channel characteristics vary (e.g., different length anddifferent gauge lines) and we have different noise and interferencecharacteristics between upstream and downstream transmission (e.g.,asymmetrical services like ADSL and VDSL; different coupling transferfunction in different directions). In this Section, we derive resultsfor neighboring lines carrying the same service when they differ inchannel, noise and interference characteristics. Specifically, wedevelop test conditions to determine the signaling nature in a given bink.

4.7.1 Solution for 2 Lines: EQPSD and FDS Signaling

Consider the case of 2 lines with different channel, noise andinterference characteristics. We again divide the channel into severalequal bandwidth bins (see FIG. 16) and continue our design and analysison one frequency bin k assuming the subchannel frequency responses(1)-(3). For ease of notation in this Section, for line 1 we setH ₁ =H _(i,k) , X ₁ =X _(i,k) , F ₁ =F _(i,k) as in (1)-(3),  (70)and letN ₁ =N _(o)(f _(k))+DS _(N)(f _(k))+DS _(F)(f _(k)),  (71)be the lumped noise PSD in line 1 bin k. Further, let P_(m1) and P_(m2)be the average powers over range [0, W] Hz in bin k of line 1 and 2respectively. Let s₁ ^(u)(f) and s₁ ^(d)(f) denote the PSDs in bin k ofline 1 upstream and downstream directions and s₂ ^(u)(f) and s₂ ^(d)(f)denote the PSDs in bin k of line 2 upstream and downstream directions(recall the notation introduced in Section 4.1, Item 9). Thecorresponding capacities of the subchannel k are denoted by c₁ ^(u), c₁^(d), c₂ ^(u) and c₂ ^(d).

We desire a signaling scheme that can have FDS, EQPSD and allcombinations in between in a frequency bin. Therefore we divide each binin half and define the upstream and downstream transmit spectra asfollows (see FIG. 41): $\begin{matrix}{{s_{1}^{u}(f)} = \left\{ \begin{matrix}{\alpha\frac{2P_{m1}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\left( {1 - \alpha} \right)\frac{2P_{m1}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix} \right.} & (72) \\{{s_{2}^{d}(f)} = \left\{ \begin{matrix}{\left( {1 - \alpha} \right)\frac{2P_{m2}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\alpha\frac{2P_{m2}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix} \right.} & (73) \\{{s_{2}^{u}(f)} = \left\{ {\begin{matrix}{\alpha\frac{2P_{m2}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\left( {1 - \alpha} \right)\frac{2P_{m2}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix}{\quad\quad{and}}} \right.} & (74) \\{{s_{1}^{d}(f)} = \left\{ \begin{matrix}{\left( {1 - \alpha} \right)\frac{2P_{m1}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\alpha\frac{2P_{m1}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix} \right.} & (75)\end{matrix}$where 0.5≦α≦1. We assume that the upstream and downstream transmitspectra obey power complementarity, i.e. line 1 puts less power whereline 2 puts more and vice versa. When α=0.5, s₁ ^(u)(f)=s₁ ^(d)(f), s₂^(u)(f)=s₂ ^(d)(f) ∀fε[0, W] (EQPSD signaling); when α=1, s₁ ^(u)(f) ands₂ ^(d)(f) are disjoint (FDS signaling). The capacities of oppositedirections are equal for each line:

-   -   c₁ ^(u)=c₁ ^(d) and c₂ ^(u)=c₂ ^(d).        The factor α controls the power distribution in the bin, and W        is the bandwidth of the bin.

Next, we show that the optimal signaling strategy uses only FDS or EQPSDin each subchannel. We also derive a test condition to determine theoptimal signaling scheme to use.

The achievable rate for one frequency bin can be written as$\begin{matrix}{{{R_{A}\left( {{s_{1}^{u}(f)},{s_{2}^{d}(f)},{s_{2}^{u}(f)}} \right)} = {\int_{0}^{W}{{\log_{2}\left\lbrack {1 + \frac{{s_{1}^{u}(f)}H_{1}}{N_{1} + {{s_{2}^{d}(f)}X_{1}} + {{s_{2}^{u}(f)}F_{1}}}} \right\rbrack}{{\mathbb{d}f}.{Thus}}}}},} & (76) \\{c_{1}^{u} = {\max\limits_{0.5 \leq \alpha \leq 1}{{R_{A}\left( {{s_{1}^{u}(f)},{s_{2}^{d}(f)},{s_{2}^{u}(f)}} \right)}.}}} & (77)\end{matrix}$We will consider the upstream capacity c₁ ^(u) expression for ouranalysis. Further, we will use R_(A) for R_(A)(s₁ ^(u)(f), s₂ ^(d)(f),s₂ ^(u)(f)) in the remainder of this Section. Substituting for the PSDsfrom (72), (73) and (74) into (76) and using (77) we get the followingexpression for the upstream capacity $\begin{matrix}{c_{1}^{u} = {\frac{W}{2}{\max\limits_{0.5 \leq \alpha \leq 1}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\frac{\alpha\quad 2P_{m1}H_{1}}{W}}{N_{1} + \frac{\left( {1 - \alpha} \right)2P_{m2}X_{1}}{W} + \frac{\alpha\quad 2P_{m2}F_{1}}{W}}} \right\rbrack} + {\log_{2}\left\lbrack {1 + \frac{\frac{\left( {1 - \alpha} \right)2P_{m1}H_{1}}{W}}{N_{1} + \frac{{\alpha 2}\quad P_{m2}X_{1}}{W} + \frac{\left( {1 - \alpha} \right)2\quad P_{m2}F_{1}}{W}}} \right\rbrack}} \right\}.}}}} & (78)\end{matrix}$Let${G_{1} = \frac{2P_{m1}}{{WN}_{1}}},\quad{{{and}\quad G_{2}} = \frac{2P_{m2}}{{WN}_{1}}}$denote the SNRs in the bin due to line 1 and line 2 respectively. Then,we can rewrite (78) as $\begin{matrix}{c_{1}^{u} = {\max\limits_{0.5 \leq \alpha \leq 1}{\frac{W}{2}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\alpha\quad G_{1}H_{1}}{1 + {\left( {1 - \alpha} \right)G_{2}X_{1}} + {\alpha\quad G_{2}F_{1}}}} \right\rbrack} + {\log_{2}\left\lbrack {1 + \frac{\left( {1 - \alpha} \right)G_{1}H_{1}}{1 + {\alpha\quad G_{2}X_{1}} + {\left( {1 - \alpha} \right)G_{2}F_{1}}}} \right\rbrack}} \right\}.}}}} & (79)\end{matrix}$

Using (77) and differentiating the achievable rate (R_(A)) expression in(79) with respect to α gives us $\begin{matrix}{{\frac{\partial R_{A}}{\partial\alpha} = {{\left( {{2\alpha} - 1} \right)\left\lbrack {{G_{2}^{2}\left( {X_{1}^{2} - F_{1}^{2}} \right)} + {2\quad{G_{2}\left( {X_{1} - F_{1}} \right)}} - {G_{1}{H_{1}\left( {{G_{2}F_{1}} + 1} \right)}}} \right\rbrack}L}},} & (80)\end{matrix}$with L>0 ∀αε(0, 1]. Setting the derivative to zero gives us the singlestationary point α=0.5. Thus, the achievable rate R_(A) is monotonic inthe interval αε(0.5, 1] (see FIG. 24). If the value α=0.5 corresponds toa maximum of R_(A), then it is optimal to perform EQPSD signaling inthis bin. If the value α=0.5 corresponds to a minimum of R_(A), then themaximum is achieved by the value α=1, meaning it is optimal to performFDS signaling in this bin. No other values of α are an optimal option.

The quantity α=0.5 corresponds to a maximum of R_(A) (EQPSD) if and onlyif $\frac{\partial R_{A}}{\partial\alpha} < 0$∀αε(0.5, 1]. For all αε(0.5, 1], $\frac{\partial R_{A}}{\partial\alpha}$is negative if and only if (see (80))G ₂ ²(X ₁ ² −F ₁ ²)+2G ₂(X ₁ −F ₁)−G ₁ H ₁(G ₂ F ₁+1)<0.This implies that $\begin{matrix}{G_{1} > {\frac{{G_{2}^{2}\left( {X_{1}^{2} - F_{1}^{2}} \right)} + {2\quad{G_{2}\left( {X_{1} - F_{1}} \right)}}}{{G_{2}F_{1}H_{1}} + H_{1}}.}} & (81)\end{matrix}$

In a similar fashion α=0.5 corresponds to a minimum of R_(A) if and onlyif $\frac{\partial R_{A}}{\partial\alpha} > 0$∀αε(0.5, 1]. This implies that α=1 corresponds to a maximum (FDS) sincethere is only one stationary point in the interval αε[0.5, 1] (see FIG.24). For all αε(0.5, 1], $\frac{\partial R_{A}}{\partial\alpha}$is positive if and only if (see (80))G ₂ ²(X ₁ ² −F ₁ ²)+2G ₂(X ₁ −F ₁)−G ₁ H ₁(G ₂ F ₁+1)>0.This implies that $\begin{matrix}{G_{1} < {\frac{{G_{2}^{2}\left( {X_{1}^{2} - F_{1}^{2}} \right)} + {2\quad{G_{2}\left( {X_{1} - F_{1}} \right)}}}{{G_{2}F_{1}H_{1}} + H_{1}}.}} & (82)\end{matrix}$

The above statements can be summed in a test condition to determine thesignaling nature (FDS or EQPSD) in a given bin. Using (81) and (82) wecan write $\begin{matrix}{G_{1} = {\frac{2\quad P_{m1}}{N_{1}W}\begin{matrix}\begin{matrix}\begin{matrix}{EQPSD} \\ > \end{matrix} \\ < \end{matrix} \\{FDS}\end{matrix}{\frac{{G_{2}^{2}\left( {X_{1}^{2} - F_{1}^{2}} \right)} + {2\quad{G_{2}\left( {X_{1} - F_{1}} \right)}}}{{G_{2}F_{1}H_{1}} + H_{1}}.}}} & (83)\end{matrix}$

Thus, we can write the upstream capacity c₁ ^(u) of line 1 in bin k as$\begin{matrix}{c_{1}^{u} = \left\{ \begin{matrix}{{W\quad{\log_{2}\left\lbrack {1 + \frac{P_{m1}H_{1}}{{N_{1}W} + {P_{m2}\left( {X_{1} + F_{1}} \right)}}} \right\rbrack}},} & {{{{if}\quad\alpha} = 0.5},} \\{{\frac{w}{2}{\log_{2}\left\lbrack {1 + \frac{2P_{m1}H_{1}}{{N_{1}W} + {2P_{m2}F_{1}}}} \right\rbrack}},} & {{{if}\quad\alpha} = 1.}\end{matrix} \right.} & (84)\end{matrix}$

4.7.2 Solution for M Lines: EQPSD and FDS Signaling

It is straightforward to generalize the result in the previous Sectionto M lines where each line i has parameters H_(i), G_(i), P_(mi), X_(i)and F_(i) for iε{1, . . . , M}. Further, we assume that the self-NEXTand self-FEXT coupling transfer functions between lines 2, . . . , M andline 1 are all the same. The test condition to determine signalingnature (EQPSD or FDS) in bin k of line 1 for M line case can be writtenas $\begin{matrix}{G_{1} = {\frac{2\quad P_{m1}}{N_{1}W}\begin{matrix}\begin{matrix}\begin{matrix}{EQPSD} \\ > \end{matrix} \\ < \end{matrix} \\{FDS}\end{matrix}{\frac{{\left( {\underset{i = 2}{\sum\limits^{M}}G_{i}} \right)^{2}\left( {X_{1}^{2} - F_{1}^{2}} \right)} + {2\left( {\underset{i = 2}{\sum\limits^{M}}G_{i}} \right)\left( {X_{1} - F_{1}} \right)}}{{\left( {\underset{i = 2}{\sum\limits^{M}}G_{i}} \right)F_{1}H_{1}} + H_{1}}.}}} & (85)\end{matrix}$

We can write the upstream capacity of line 1 in bin k as $\begin{matrix}{c_{1}^{u} = \left\{ \begin{matrix}{{W\quad{\log_{2}\left\lbrack {1 + \frac{P_{m1}H_{1}}{{N_{1}W} + {\left( {\underset{i = 2}{\sum\limits^{M}}P_{m\quad i}} \right)\left( {X_{1} + F_{1}} \right)}}} \right\rbrack}},} & {{{{if}\quad\alpha} = 0.5},} \\{{\frac{W}{2}{\log_{2}\left\lbrack {1 + \frac{2P_{m1}H_{1}}{{N_{1}W} + {2\left( {\underset{i = 2}{\sum\limits^{M}}P_{m\quad i}} \right)F_{1}}}} \right\rbrack}},} & {{{if}\quad\alpha} = 1.}\end{matrix} \right.} & (86)\end{matrix}$

4.7.3 Solution for 2 Lines: EQPSD and Multi-line FDS Signaling

We saw in Section 4.6.8 that in the case of two lines it is optimal touse multi-line FDS instead of FDS signaling. In this Section we willderive a test condition to determine the signaling nature in a givenbin. We use the notation as introduced in Section 4.7.1.

We desire a signaling scheme that supports multi-line FDS, EQPSD, andall combinations in between in a frequency bin. Therefore we divide eachbin in half and define the upstream and downstream transmit spectra asfollows (see FIG. 42): $\begin{matrix}{{s_{1}^{u}(f)} = {{s_{1}^{d}(f)} = \left\{ \begin{matrix}{\alpha\frac{2P_{m1}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\left( {1 - \alpha} \right)\frac{2P_{m1}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix} \right.}} & (87) \\{{s_{2}^{u}(f)} = {{s_{2}^{d}(f)} = \left\{ \begin{matrix}{\left( {1 - \alpha} \right)\frac{2P_{m2}}{W}} & {{{{if}\quad{f}} \leq \frac{W}{2}},} \\{\alpha\frac{2P_{m2}}{W}} & {{{{if}\quad\frac{W}{2}} < {f} \leq W},} \\0 & {{otherwise},}\end{matrix} \right.}} & (88)\end{matrix}$where 0.5≦α≦1. We assume that the upstream and downstream transmitspectra obey power complementarity, i.e., line 1 puts less power whereline 2 puts more and vice versa. In further discussion we will usetransmit spectra s₁ ^(u)(f) and s₂ ^(d)(f). When α=0.5, s₁ ^(u)(f)=s₂^(d)(f), ∀fε[0, W] (EQPSD signaling); when α=1, s₁ ^(u)(f) and s₂^(d)(f) are disjoint (FDS signaling). The capacities of oppositedirections are equal for each line:

-   -   c₁ ^(u)=c₁ ^(d) and c₂ ^(u)=c₂ ^(d).        The factor α controls the power distribution in the bin and W is        the bandwidth of the bin.

Next, we show that the optimal signaling strategy uses only EQPSD ormulti-line FDS in each subchannel and derive a test condition todetermine the signaling scheme to use.

The achievable rate for one frequency bin can be written as$\begin{matrix}{{{R_{A}\left( {{s_{1}^{u}(f)},{s_{2}^{d}(f)},{s_{2}^{u}(f)}} \right)} = {\int_{0}^{W}{{\log_{2}\left\lbrack {1 + \frac{{s_{1}^{u}(f)}H_{1}}{N_{1} + {{s_{2}^{d}(f)}X_{1}} + {{s_{2}^{u}(f)}F_{1}}}} \right\rbrack}{\mathbb{d}f}}}},{then}} & (89) \\{c_{1}^{u} = {\max\limits_{0.5 \leq \alpha \leq 1}{{R_{A}\left( {{s_{1}^{u}(f)},{s_{2}^{d}(f)},{s_{2}^{u}(f)}} \right)}.}}} & (90)\end{matrix}$We will consider the upstream capacity c₁ ^(u) expression for ouranalysis. Further, we will use R_(A) for R_(A)(s₁ ^(u)(f), s₂ ^(d)(f),s₂ ^(u)(f)) in the remainder of this Section. Substituting for the PSDsfrom (72) and (73) into (89) and using (90) we get the followingexpression for the upstream capacity $\begin{matrix}{c_{1}^{u} = {\frac{W}{2}{\max\limits_{0.5 \leq \alpha \leq 1}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\frac{\alpha\quad 2P_{m1}H_{1}}{W}}{N_{1} + \frac{\left( {1 - \alpha} \right)2\quad P_{m2}X_{1}}{W} + \frac{\left( {1 - \alpha} \right)2P_{m2}F_{1}}{W}}} \right\rbrack} + {\log_{2}\left\lbrack \quad{1 + \frac{\frac{\left( {1 - \alpha} \right)\quad 2P_{m1}H_{1}}{W}}{N_{1} + \frac{\alpha\quad 2P_{m2}X_{1}}{W} + \frac{\alpha\quad 2P_{m2}F_{1}}{W}}} \right\rbrack}} \right\}.}}}} & (91)\end{matrix}$Let${G_{1} = \frac{2P_{m1}}{{WN}_{1}}},\quad{{{and}\quad G_{2}} = \frac{2P_{m2}}{{WN}_{1}}}$denote the SNRs in the bin due to line 1 and line 2 respectively. Then,we can rewrite (91) as $\begin{matrix}{c_{1}^{u} = {\max\limits_{0.5 \leq \alpha \leq 1}{\frac{W}{2}{\left\{ {{\log_{2}\left\lbrack {1 + \frac{\alpha\quad G_{1}H_{1}}{1 + {\left( {1 - \alpha} \right)G_{2}X_{1}} + {\left( {1 - \alpha} \right)G_{2}F_{1}}}} \right\rbrack} + {\log_{2}\left\lbrack {1 + \frac{\left( {1 - \alpha} \right)\quad G_{1}H_{1}}{1 + {\alpha\quad G_{2}X_{1}} + {\alpha\quad G_{2}F_{1}}}} \right\rbrack}} \right\}.}}}} & (92)\end{matrix}$

Using (90) and differentiating the achievable rate (R_(A)) expression in(92) with respect to α gives us $\begin{matrix}{{\frac{\partial R_{A}}{\partial\alpha} = {{\left( {{2\alpha} - 1} \right)\left\lbrack {{G_{2}^{2}\left( {X_{1} + F_{1}} \right)}^{2} + {2{G_{2}\left( {X_{1} + F_{1}} \right)}} - {G_{1}H_{1}}} \right\rbrack}L}},} & (93)\end{matrix}$with L>0 ∀αε(0, 1]. Setting the derivative to zero gives us the singlestationary point α=0.5. Thus, the achievable rate R_(A) is monotonic inthe interval αε(0.5, 1] (see FIG. 24). If the value α=0.5 corresponds toa maximum of R_(A), then it is optimal to perform EQPSD signaling inthis bin. If the value α=0.5 corresponds to a minimum of R_(A), then themaximum is achieved by the value α=1, meaning it is optimal to performmulti-line FDS signaling in this bin. No other values of α are anoptimal option.

The quantity α=0.5 corresponds to a maximum of R_(A) (EQPSD) if and onlyif $\frac{\partial R_{A}}{\partial\alpha} < 0$∀αε(0.5, 1]. For all αε(0.5, 1], $\frac{\partial R_{A}}{\partial\alpha}$is negative if and only if (see (93))G ₂ ²(X ₁ +F ₁)²+2G ₂(X ₁ +F ₁)−G ₁ H ₁<0.This implies that $\begin{matrix}{G_{1} > {\frac{{G_{2}^{2}\left( {X_{1} + F_{1}} \right)}^{2} + {2{G_{2}\left( {X_{1} + F_{1}} \right)}}}{H_{1}}.}} & (94)\end{matrix}$

In a similar fashion α=0.5 corresponds to a minimum of R_(A) if and onlyif ${\frac{\partial R_{A}}{\partial\alpha} > 0}\quad$∀αε(0.5, 1]. This implies that α=1 corresponds to a maximum of R_(A)(multi-line FDS) since there is only one stationary point in theinterval αε[0.5, 1] (see FIG. 24). For all αε(0.5, 1],$\frac{\partial R_{A}}{\partial\alpha}$is positive if and only if (see (93))G ₂ ²(X ₁ +F ₁)²+2G ₂(X ₁ +F ₁)−G ₁ H ₁>0.This implies that $\begin{matrix}{G_{1} < {\frac{{G_{2}^{2}\left( {X_{1} + F_{1}} \right)}^{2} + {2{G_{2}\left( {X_{1} + F_{1}} \right)}}}{H_{1}}.}} & (95)\end{matrix}$

The above statements can be summed in a test condition to determine thesignaling nature (EQPSD or multi-line FDS) in a given bin. Using (94)and (95) we can write $\begin{matrix}{G_{1} = {\frac{2P_{m1}}{N_{1}W}\quad\begin{matrix}{EQPSD} \\ > \\ < \\{{multi}\text{-}{lineFDS}}\end{matrix}{\frac{{G_{2}^{2}\left( {X_{1} + F_{1}} \right)^{2}} + {2{G_{2}\left( {X_{1} + F_{1}} \right)}}}{H_{1}}.}}} & (96)\end{matrix}$

Thus, we can write the upstream capacity c₁ ^(u) of line 1 in bin k as$\begin{matrix}{c_{1}^{u} = \left\{ \begin{matrix}{{W\quad{\log_{2}\left\lbrack {1 + \frac{P_{m1}H_{1}}{{N_{1}W} + {P_{m2}\left( {X_{1} + F_{1}} \right)}}} \right\rbrack}},} & {{{{if}\quad\alpha} = 0.5},} \\{{\frac{W}{2}\quad{\log_{2}\left\lbrack {1 + \frac{2P_{m1}H_{1}}{N_{1}W}} \right\rbrack}},} & {{{if}\quad\alpha} = 1.}\end{matrix} \right.} & (97)\end{matrix}$4.8 Optimizing Under a PSD Mask Constraint: No Self-interference

In this Section we will impose an additional peak power constraint infrequency, i.e., a limiting static PSD mask constraint. This impliesthat no transmit spectrum can lie above the PSD mask constraint. Thisconstraint is in addition to the average power constraint. We shallobtain optimal transmit spectra for an xDSL line under theseconstraints, in the absence of self-interference.

4.8.1 Problem Statement

Maximize the capacity of an xDSL line in the presence of AGN andinterference (DSIN-NEXT and DSIN-FEXT) from other services under twoconstraints:

-   -   1. The xDSL transmit spectra are limited by constraining static        PSD masks; Q^(u)(f) for upstream and Q^(d)(f) for downstream.    -   2. The average xDSL input power in each direction of        transmission must be limited to P_(max) (Watts).        Do this by designing the distribution of energy over frequency        (the transmit spectrum) of the xDSL transmission.

4.8.2 Solution

Consider a line (line 1) carrying an xDSL service. Line 1 experiencesinterference from other neighboring services (DSIN-NEXT and DSIN-FEXT)and channel noise N_(o)(f) (AGN) but no self-NEXT or self-FEXT (see FIG.19).

The twisted pair channel can be treated as a Gaussian channel withcolored Gaussian noise [13]. Recall that DS_(N)(f) is the PSD of thecombined DSIN-NEXT and DS_(F)(f) is the PSD of the combined DSIN-FEXT.Let S^(u)(f) and S^(d)(f) denote the PSDs of line 1 upstream (u)direction and downstream (d) direction transmitted signals,respectively. Further, let C^(u) and C^(d) denote the upstream anddownstream direction capacities of line 1 respectively. Let H_(C)(f)denote the channel transfer function of line 1.

The channel capacities (in bps) are given by [14] $\begin{matrix}{C^{u} = {\sup\limits_{S^{u}{(f)}}{\int_{0}^{\infty}{{\log_{2}\quad\left\lbrack {1 + \frac{{{{H_{C}(f)}}^{2}{S^{u}(f)}}\quad}{{N_{o}(f)} + {D\quad{S_{N}(f)}} + {D\quad{S_{F}(f)}}}} \right\rbrack}{\mathbb{d}f}\quad{and}}}}} & (98) \\{C^{d} = {\sup\limits_{S^{d}{(f)}}{\int_{0}^{\infty}{{\log_{2}\quad\left\lbrack {1 + \frac{{{{H_{C}(f)}}^{2}{S^{d}(f)}}\quad}{\left. {N_{o}(f)} \right) + {D\quad{S_{N}(f)}} + {D\quad{S_{F}(f)}}}} \right\rbrack}{{\mathbb{d}f}.}}}}} & (99)\end{matrix}$The supremum is taken over all possible S^(u)(f) and S^(d)(f) satisfyingthe average power constraints for the two directions2∫₀ ^(∞) S ^(u)(f)df≦P _(max) and 2∫₀ ^(∞) S ^(d)(f)df≦P _(max),  (100)and the positivity and new peak power constraints0≦S ^(u)(f)≦Q ^(u)(f) ∀f and 0≦S ^(d)(f)≦Q ^(d)(f) ∀f,  (101)Note that these equations are the same as (4)-(6) except for theadditional peak power constraint in frequency. For discussion purposes,we will focus on the upstream transmission. The same analysis can beapplied to the downstream channel.

We wish to maximize (98) subject to the constraints (100), (101). Theconstraints (100), (101) are differentiable and concave. Further, theobjective function to be maximized (98) is also concave (the logfunction is concave). Any solution to this problem must satisfy thenecessary KKT (Karush-Kuhn-Tucker) [22] conditions for optimality. For aconcave objective function and concave, differentiable constraints; anysolution that satisfies the necessary KKT conditions is a uniqueglobally optimal solution [22]. Thus, we seek any solution thatsatisfies the KKT conditions, since it is automatically the uniqueoptimal solution.

The optimal solution to (98), (99), (100), (101) is basically a“peak-constrained water-filling”.⁶ The optimal transmit spectrum isgiven by $\begin{matrix}{{S_{opt}^{u}(f)} = \left\{ \begin{matrix}{\lambda - \frac{\left. {N_{o}(f)} \right) + {D\quad{S_{N}(f)}} + {D\quad{S_{F}(f)}}}{{{H_{C}(f)}}^{2}}} & {{{{for}\quad f} \in E_{pos}},} \\{Q^{u}(f)} & {{{{for}\quad f} \in E_{\max}},} \\0 & {{otherwise},}\end{matrix} \right.} & (102)\end{matrix}$with λ a Lagrange multiplier. The spectral regions E_(pos) and E_(max)are specified byE _(pos) ={f:0≦S ^(u)(f)≦Q ^(u)(f)}, andE _(max) ={f:S ^(u)(f)>Q ^(u)(f)}.  (103)We vary the value of λ to achieve the optimal transmit spectrum S_(opt)^(u)(f) that satisfies the average and peak power constraints (100),(101). It can be easily shown that this solution satisfies the KKTconditions for optimality. Substituting the optimal PSD S_(opt) ^(u)(f)into (98) yields the capacity C^(u) under the average and peak powerconstraints.

⁶Peak-constrained water-filling can be likened to filling water in aclosed vessel with uneven top and bottom surfaces.

Note that if the maximum allowed average power (P_(max)) exceeds thepower under the constraining mask then the optimal transmit spectrum isthe constraining PSD mask itself. In the absence of an average powerconstraint (but with a peak power constraint) the optimal transmitspectrum is again the constraining PSD mask.

4.8.3 Examples

In this Section we consider a line carrying HDSL2 service under theOPTIS [5] constraining PSD mask and input power specifications. Anaverage input power (P_(max)) of 19.78 dBm and a fixed bit rate of 1.552Mbps was used for all simulations.

FIG. 43 shows the optimal downstream transmit spectrum for HDSL2 withOPTIS downstream constraining mask in the presence of DSIN-NEXT from 49HDSL interferers and AGN (−140 dBm/Hz). The key features in the case ofHDSL interferers are:

-   -   1. Comparing the peak-constrained transmit spectrum in FIG. 43        with the unconstrained in peak power one in FIG. 21 indicates        that the peak-constrained optimal solution tries to follow the        unconstrained in peak power optimal solution. The        peak-constrained optimal solution has a null in the spectrum        around 150 kHz similar to the one in the unconstrained in peak        power spectrum. The null in the transmit spectra occurs in order        to avoid the interfering HDSL transmit spectrum.    -   2. An OPTIS transmit spectrum, achieved by tracking 1 dBm/Hz        below the OPTIS PSD mask throughout, does not yield good        performance margins (see Table 7). The OPTIS transmit spectrum        looks different from the peak-constrained optimal spectrum (see        FIG. 43). The null in the peak-constrained optimal spectrum        (which is not seen in the OPTIS transmit spectrum) indicates        that it is suboptimal to distribute power according to the OPTIS        transmit spectrum.

TABLE 7 Uncoded performance margins (in dB) for CSA No. 6: OPTIS vs.Peak-constrained Optimal “under OPTIS” xDSL OPTIS Optimal Diff CrosstalkSrc service Dn Up Dn Up Dn Up 49 HDSL HDSL2 12.24 2.7 13.74 3.74 1.541.03 25 T1 HDSL2 17.5 19.9 18.81 20.43 1.31 0.53 39 self HDSL2 9.0 2.115.51 17.58 6.51 15.48 24 self + 24 T1 HDSL2 1.7 4.3 4.74 4.52 3.04 0.22Bit rate fixed at 1.552 Mbps. Average Input power = 19.78 dBm. Diff (Dn)= Difference in Downstream margins (Optimal − OPTIS) Diff (Up) =Difference in Upstream margins (Optimal − OPTIS)

FIG. 44 shows the optimal upstream transmit spectrum for HDSL2 withOPTIS upstream constraining mask in the presence of DSIN-NEXT from 25 T1interferers and AGN (−140 dBm/Hz). Again, we compare thepeak-constrained transmit spectrum in FIG. 44 with the unconstrained inpeak power one in FIG. 22. Note that the peak-constrained optimaltransmit spectrum puts no power in the high-frequency spectrum (to avoidT1 interference) as opposed to an OPTIS transmit spectrum.

4.9 Optimizing Under a PSD Mask Constraint: With Self-interference

The solution outlined in the previous Section applies only in theabsence of self-interference. In this Section we will find an optimaltransmit spectrum in the presence of additional self-NEXT and self-FEXT.We will impose a peak power constraint in frequency, i.e., a limitingstatic PSD mask constraint, in addition to the average power andsymmetric bit-rate constraints. We will obtain the optimal transmitspectra for an xDSL line under these constraints in the presence ofself-interference.

4.9.1 Problem Statement

Maximize the capacity of an xDSL line in the presence of AGN,interference (DSIN-NEXT and DSIN-FEXT) from other services, andself-interference (self-NEXT and self-FEXT) under three constraints:

-   -   1. The xDSL transmit spectra are limited by constraining static        PSD masks; Q^(u)(f) for upstream and Q^(d)(f) for downstream.    -   2. The average xDSL input power in each direction of        transmission must be limited to P_(max) (Watts).    -   3. Equal capacity in both directions (upstream and downstream)        for xDSL.        Do this by designing the distribution of energy over frequency        (the transmit Spectra) of the xDSL transmissions.

Additional assumptions are made in this case as given in Section 4.5.3or 4.6.3 depending on the signaling scheme used.

4.9.2 Solution

Consider a line (line 1) carrying xDSL service. Line 1 experiencesinterference from other neighboring services (DSIN-NEXT and DSIN-FEXT),channel noise N_(o)(f) (AGN), and self-interference (self-NEXT andself-FEXT) (see FIG. 3).

We need to find peak-constrained optimal transmit spectra for upstreamand downstream transmission. We let the constraining PSD mask Q(f) bethe maximum of the two upstream and downstream constraining masks(Q^(u)(f) and Q^(d)(f)) We then employ the solutions as described inSections 4.5 or 4.6 but limit the peak power to the constraining maskQ(f). Thus, we obtain a peak-constrained transmit spectrum S_(opt)(f).Using this mask, we optimally group the bins (see Section 4.5.10) toobtain optimal upstream and downstream transmit spectra (S₁ ^(u)(f) andS₁ ^(d)(f)).

4.9.3 Algorithm for Peak-constrained Optimization of the TransmitSpectra

-   -   1. Choose the constraining PSD mask as        Q(f)=max(Q ^(u)(f), Q ^(d)(f)) ∀f.    -   2. Solve for the optimal transmit spectrum S_(opt) ^(u)(f)        according to the algorithms in Sections 4.5.7, 4.5.8, or 4.6        with the following added constraint $\begin{matrix}        {{S_{opt}^{u}(f)} = \left\{ \begin{matrix}        {Q(f)} & {{\forall{{f\quad{where}\quad{S^{u}(f)}} > {Q(f)}}},} \\        {S^{u}(f)} & {{otherwise},}        \end{matrix} \right.} & (104)        \end{matrix}$    -    where S^(u)(f) is the water-filling solution (refer to [14] if        the spectral region employs EQPSD or multi-line FDS signaling        and to [16] if the spectral region employs FDS signaling) (see        Sections 4.5 and 4.6). This is the peak-constrained        water-filling solution in the presence of self-interference. As        argued in the previous Section, this solution satisfies the        necessary KKT conditions for optimality and therefore is the        unique optimal solution.    -   3. Denote the spectral region employing FDS signaling as EFDS        and the spectral region employing EQPSD signaling as E_(EQPSD).    -    Obtain S_(opt) ^(d)(f) from S_(opt) ^(u)(f) by symmetry, i.e.,        S_(opt) ^(d)(f)=S_(opt) ^(u)(f) in EQPSD and multi-line FDS        regions and S_(opt) ^(d)(f)⊥S_(opt) ^(u)(f) in FDS spectral        regions. Merge S_(opt) ^(d)(f) and S_(opt) ^(u)(f) to form        S_(opt)(f) as        S _(opt)(f)=S _(opt) ^(u)(f)=S _(opt) ^(d)(f) ∀f in E _(EQPSD),        S _(opt)(f)=S _(opt) ^(u)(f)∪S _(opt) ^(d)(f) ∀f in E        _(FDS),  (105)    -    where ∪ represents the union of the two transmit spectra    -    Group the bins to obtain upstream and downstream masks as        S ₁ ^(u)(f)=S _(opt)(f) ∀f in E _(FDS) and where Q ^(u)(f)≧Q        ^(d)(f),        S ₁ ^(d)(f)=S _(opt)(f) ∀f in E _(FDS) and where Q ^(u)(f)<Q        ^(d)(f)  (106)    -    in EFDS and        S ₁ ^(u)(f)=S ₁ ^(d)(f)=S _(opt)(f) ∀f in E _(EQPSD).  (107)    -   4. Check if the average power constraint is violated for        upstream or downstream transmission.    -   5. If the average power constraint is violated for direction o        (i.e., the total transmit power in the direction o is more than        P_(max))⁷ then transfer power from S₁ ^(o)(f) to S₁ ^(ō)(f).        Transfer power first from spectral regions of S₁ ^(o)(f) to S₁        ^(ō)(f) with the least S₁ ^(o)(f)−S₁ ^(ō)(f) difference. Repeat        this successively in spectral regions with increasing S₁        ^(o)(f)−S₁ ^(ō)(f) difference until the average power in both        directions is the same.⁸ We transfer power from one direction o        to the other direction ō in spectral regions where the        difference in power between the two transmission directions is        the least until the power between the two directions becomes        equal. This power transfer scheme is in a sense optimal as it        tries to even out the powers between the two directions, with        the least loss in the total sum of the transmit powers of the        two directions.    -    If the difference S₁ ^(o)(f)−S₁ ^(ō)(f) is the same (or        marginally varying) for a range of frequencies, then transfer        power from direction o to direction ō in those spectral regions        that give the maximum gain in bit rates for direction ō.

⁷Note that if the total transmit power in direction o is more thanP_(max) then the transmit power in direction ō is less than P_(max).

⁸This approach of transferring power from direction o to direction ō canbe likened to “stealing from the rich and giving to the poor.”

4.9.4 Examples and Results

In this Section we consider a line carrying HDSL2 service under theOPTIS [5] constraining PSD mask and input power specifications. Anaverage input power (P_(max)) of 19.78 dBm and a fixed bit rate of 1.552Mbps was used for all simulations.

FIGS. 45A and 45B show the optimal upstream and downstream transmitspectra for HDSL2 with OPTIS constraining masks in the presence ofself-NEXT and self-FEXT from 39 HDSL2 interferes and AGN (−140 dBm/Hz).Note that the optimal upstream and downstream transmit spectra areseparated in frequency (using FDS signaling) in a large spectral regionin order to avoid high self-NEXT. On the other hand, OPTIS transmitspectra have a large spectral overlap at lower frequencies (self-NEXT ishigh here) that significantly reduces its performance margins (see Table7).

FIGS. 46A and 46B show the optimal upstream and downstream transmitspectra for HDSL2 with OPTIS constraining masks in the presence ofself-NEXT and self-FEXT from 24 HDSL2 interferes, DSIN-NEXT from 24 T1interferers, and AGN (−140 dBm/Hz). Again, we see that the upstream anddownstream optimal spectra are separated in frequency (using FDSsignaling) over a large spectral region. However, the EQPSD spectralregion towards the beginning of the spectrum is larger here than in theprevious example, since we have more DSIN-NEXT from T1.

Key here is that optimal transmit spectra employ optimal separation infrequency of upstream and downstream services in the presence ofinterference. The “1 dB below OPTIS” transmit spectra do not do this,and so have inferior performance.

Table 7 compares the performance margins of the OPTIS transmit spectra(obtained from the OPTIS PSD mask by uniformly subtracting 1 dBm/Hz overthe entire frequency range as in [5]) with the optimal transmit spectraunder the OPTIS PSD mask constraints. Table 7 shows that the optimalscheme significantly outperforms OPTIS in the case of self-interference.In cases involving different service interferers (HDSL and T1) theoptimal scheme consistently outperforms OPTIS by 1 dB or more. Further,comparing these results with those in Table 1 suggests that the OPTISPSD mask is not a good constraining PSD mask, since the unconstrained inpeak power margins in Table 1 are significantly higher than the ones inTable 7. Comparing Tables 1 and 7 suggests that optimal signaling withno peak power constraint (static PSD mask) gives high performance margingains.

4.10 Bridged Taps

Bridged taps (BTs) are shorter segments of twisted pairs that attach toanother twisted pair that carries data between the subscriber and theCO. BTs are terminated at the other end with some characteristicimpedance. BTs reflect the signals on the data-carrying line. Thesereflections destructively interfere with the transmitted signal overcertain frequencies. This leads to nulls in the channel transferfunction and the self-NEXT transfer function at these frequencies (seeFIGS. 48A and 48B). These nulls in the channel transfer functionsignificantly reduce the data transmission rate. Thus, bridged taps posean important problem in achieving high bit rates over xDSL lines.⁹

⁹Bridged taps can be removed from xDSL lines, but this is an expensive(labor-intensive) procedure.

Bridged taps presence, location, and length vary according to each loopsetup. Thus, the effect of BTs on the transmission signals is differentfor each loop. This means that the channel transfer function nulls (infrequency) vary for each separate line. We need to adapt the transmitspectrum to the channel conditions in order to achieve high bit-rates.We need the optimal power distribution that maximizes the bit-rates inthe presence of bridged taps and interference. This further enforces theneed for optimal dynamic transmit spectra and indicates that statictransmit spectra are not a good idea. In this Section, we presentoptimal and near-optimal solutions to find the transmit spectra in thepresence of BTs.

4.10.1 Optimal Transmit Spectra

Optimal signaling is more computationally expensive to implement in thepresence of bridged taps [3), as the channel transfer function has nullsand thus loses its monotonicity. In this scenario, even the self-FEXTtransfer function has nulls. In spite of this, the overall optimalsolution can be obtained by a bin by bin analysis:

-   -   1. Divide the frequency axis into narrow bins or subchannels.        Compute channel transfer function, various interference transfer        functions, and AGN.    -   2. Choose an initial power distribution of P_(max) over all        bins.    -   3. Given the powers in each bin decide the optimal signaling        scheme in each bin. Compute capacities for each bin and hence        compute channel capacity.    -   4. Re-distribute the powers in each bin by water-filling [14],        [16], decide the optimal signaling scheme in each bin, and        re-calculate the channel capacity. Repeat this step until we        find the maximum possible channel capacity. It can be        exceedingly computationally intensive to find the optimal power        distribution over all bins. There can be several local maxima        for the channel capacity curve, and there is no guarantee that a        search algorithm will converge to the global maximum.

The optimal power distribution algorithm suggests that EQPSD, FDS, andmulti-line FDS bins could be randomly distributed throughout thetransmission bandwidth. The search for the optimal switchover bins fromone signaling scheme to the other could be exceedingly expensive(involving a multi-dimensional search).

4.10.2 Suboptimal Transit Spectra

We saw in the previous Section that the optimal transmit spectrum couldbe very expensive to obtain. However, we can always get a goodsuboptimal solution for line i as follows:

-   -   1. Divide the frequency axis into narrow bins or subchannels as        in Section 4.1. Compute channel transfer function (H_(C)(f)),        the various interference transfer functions (H_(N)(f), H_(F)(f),        DS_(N)(f), and DS_(F)(f)), and AGN (N_(o)(f)). Obtain subchannel        values (H_(i,k), X_(i,k), F_(i,k)) for each bin using (1)-(3)        and (13). Let k denote the bin number.    -   2. Use the condition evaluations in (26) and (27) to determine        the signaling scheme (EQPSD or FDS) in each bin. For each bin:        -   If (X_(i,k) ²−F_(i,k) ²−H_(i,k)F_(i,k)<0) and the right side            of (26)<0, then employ EQPSD signaling in that bin (since            power in every bin≧0).        -   If (X_(i,k) ²−F_(i,k) ²−H_(i,k)F_(i,k)>0) and the right side            of (27)<0, then employ FDS signaling in that bin (since            power in every bin≧0).        -   Employ FDS signaling if both the above conditions are not            satisfied.    -   3. Perform the optimal power distribution under average power        constraint of P_(max) using water-filling technique [14], [16].    -   4. Use condition evaluations in (46) and (54) to determine bins        employing multi-line FDS. Redistribute power optimally using        water-filling technique. This step is optional and indicates        which bins employ multi-line FDS signaling.

The suboptimal solution determines the signaling strategy in each bin bysimple, fast comparisons involving transfer functions and SNRs. This isfollowed by a simple optimal power distribution scheme using thewater-filling technique.

Note that the optimal and suboptimal algorithms can be implemented undera peak frequency-domain power constraint (static PSD mask). This isachieved by using peak-constrained water-filling technique (instead ofjust water-filling) for optimal power distribution (see Sections 4.8 and4.9) in the algorithms given in Sections 4.10.1 and 4.10.2.

4.10.3 Examples and Discussion

Optimal transmit spectra: Theoretically, the optimal transmit spectrumin the presence of BTs can have several switchover bins from onesignaling scheme to the other (for e.g., EQPSD to FDS and FDS to EQPSDswitchover bins). However, we argue that in most of the symmetricaldata-rate services (like HDSL2 and “VDSL2”) there is only one switchoverbin from EQPSD to FDS in spite of bridged taps.

As frequency increases, the self-NEXT transfer function rapidlyincreases but the self-FEXT and the channel transfer functions generallydecrease even for bridged taps case (see FIGS. 17 and 48). Thus, thequantity X_(i,k) ²−F_(i,k)−H_(i,k)F_(i,k) tends to be an increasingfunction of frequency or bin number k, and stays positive once itbecomes positive. Similarly, the quantity H_(i,k)−2(X_(i,k)−F_(i,k))tends to decrease with frequency or bin number k and stays negative onceit becomes negative. Using the condition evaluations (26) and (27) forall the frequency bins indicate that there is only one EQPSD to FDSswitchover bin. Our studies indicate that is indeed true for a widerange of loops having bridged taps and employing HDSL2, “VDSL2” orsimilar symmetric services. The optimal switchover bin along with theoptimal transmit spectrum can be determined using the algorithm inSection 4.5.7.

FIGS. 47A and 47B illustrate a case of “contiguous” optimal transmitspectra in the case of a loop with bridged taps (CSA loop 4). We canclearly see that the optimal transmit spectra have only one transitionregion from EQPSD to FDS signaling. The transmit spectra were obtainedsuch that we have equal performance margins and equal average powers inboth directions of transmission.

Suboptimal transmit spectra: We presented strong arguments in support ofonly one EQPSD to FDS switchover bin in the previous paragraph. However,there can be exceptions when the arguments do not hold, and we havemultiple EQPSD and FDS regions (see FIGS. 48A and 48B). Consider ahypothetical case of a short loop (1.4k with 3 bridged taps) carryingthe “GDSL” service. The channel transfer function, self-NEXT, andself-FEXT transfer functions are illustrated in FIG. 48A. Note that for“GDSL” service the self-NEXT is assumed very low. Since the self-NEXT islow, the monotonicity of the self-FEXT and the channel transfer functionlead to distributed EQPSD and FDS regions across the transmissionbandwidth as illustrated in FIG. 48B. In such a scenario, the optimalpower distribution algorithm of Section 4.10.1 is exceedingly difficultto implement. However, we can easily implement the suboptimal solutionas given in 4.10.2.

4.11 Optimization: Asymmetrical Data-rate Channels

Asymmetrical data-rate channels have different upstream and downstreamtransmission rates, for e.g., ADSL and VDSL services. These channelsalso employ different average powers in the two transmission directions.We find joint signaling strategies and optimal power distribution forthese channels using similar approaches as described in previousSections (see Sections 4.5, and 4.6). In this case we assume theknowledge of the ratio of average powers between upstream and downstreamdirections.

4.12 Extensions

4.12.1 More General Signaling Techniques

The signaling techniques outlined earlier are not the only techniquesthat can give us improved capacity results. One possible scheme isillustrated in FIG. 49. In this Figure, UP_(i) and DOWN_(i) refer toline i, upstream and downstream direction PSDs respectively. In thisscheme, we use multi-line FDS between group of lines (1 and 2) havinghigh self-NEXT and high self-FEXT with other group of lines (3 and 4).However, there is EQPSD among group of lines (1 and 2 employ EQPSD as do3 and 4) that have low self-NEXT and low self-FEXT within the group.This scheme can be extended for M self-interfering lines (with differentself-NEXT and self-FEXT combinations between them) using combination ofEQPSD, FDS, and multi-line EDS signaling schemes between different linesand frequency bins.

The above scheme can be applied in the case of groups of lines withdifferent self-interference (self-NEXT and self-FEXT) characteristicsbetween different set of lines.

4.12.2 More General Interferer Models

If the self-NEXT and self-FEXT interferer model cannot be easilycharacterized by monotonicity in regions, (that is, if they vary rapidlyand non-monotonously from one subchannel to the other), then we mustsearch for the overall optimal solution on a bin by bin basis. Thissearch is outlined in the Section 4.10 on bridged taps.

4.12.3 Channel Variations

Some channels (e.g., the geophysical well-logging wireline channel)undergo a significant change in channel transfer function H_(C)(f) as afunction of temperature. Temperature variations are a part of nature andhence we need to continuously update our channel transfer functions.Changes in channel characteristics can change the channel capacity. Wecan develop an adaptive optimal transmit spectrum to adjust to these aswell as any other variations.

4.12.4 Broadband Modulation Schemes

We saw in Section 4.5.10 that we can easily group the bins of theoptimal transmit spectrum to make it smoother (with fewerdiscontinuities), so that we could apply different broadband modulationschemes. One can apply different broadband modulation schemes (likemulti-level PAM, QAM, CAP, etc.) over large spectral regions to theoptimal transmit spectrum obtained after grouping the bins and determinethe performance margins. In this case, we need to use a DFE at thereceiver to compensate for the severe channel attenuationcharacteristics. All these broadband modulation schemes do not sufferfrom latency as DMT does, but the DFE structure is complex. It isworth-while to compare the margins obtained with broadband modulationschemes with those obtained using DMT as well as compare the complexityand implementation issues involved.

4.12.5 Linear Power Constraints in Frequency

We saw in earlier Sections 4.4-4.10, optimal power distribution usingwater-filling technique under an average power constraint, andpeak-constrained water-filling technique under a peak power constraintin frequency or average plus peak power constraint in frequency. Ingeneral, we can determine the optimal power distribution under any setof general linear power constraints in frequency. Further, we can employone of the joint signaling techniques discussed in this document underthese new constraints using similar analysis.

4.12.6 CDS Signaling Under a Peak Power Constraint in Frequency

In case of a limiting static PSD mask, (see Sections 4.8 and 4.9), orotherwise, one may be required to limit the peak power in one or all thefrequency bins. In this case a power-peaky signaling scheme like FDS ormulti-line FDS will no longer be optimal as now we have a peak powerconstraint instead of the average power constraint. For this case, CDSor multi-line CDS signaling [20] would be a better orthogonal signalingtechnique and would give increased capacity benefits withoutcompromising spectral compatibility.

Recall, that in frequency bins where self-NEXT is high and self-FEXT islow, we need orthogonal signaling (FDS, TDS, or CDS) between upstreamand downstream transmissions, i.e.,S _(i) ^(u)(f)⊥s _(j) ^(d)(f), ∀i≠j.  (108)Under an average power constraint, FDS signaling is the optimalsignaling strategy (see Section 4.5.12). In FDS signaling s₁ ^(u)(f) ands_(j) ^(d)(f) occupy distinct separate frequency bands that are twice ashigher than those using EQPSD signaling (see FIG. 25). In CDS signalingthe transmit spectra s_(i) ^(u)(f) and s_(j) ^(d)(f) look similar toEQPSD signaling but the upstream and downstream spectra are separatedusing two orthogonal codes. Under a peak power constraint in frequencyCDS signaling is preferred. Towards this end, we can group together binsusing FDS into one spectral region E_(FDS). We can implement spreadspectrum CDS (SS-CDS) over this spectral region E_(FDS) such thatS _(i) ^(u)(f)⊥S _(j) ^(d)(f), ∀i≠j, ∀fεE _(FDS)  (109)

Further, recall that in frequency bins where self-FEXT is high, we needto use orthogonal signaling (multi-line FDS, TDS, or multi-line CDS)between upstream and downstream transmissions of all the M lines, i.e.,s _(i) ^(o)(f)⊥s _(j) ^(d)(f), ∀i≠j, oε{u,d}  (110)Multi-line CDS separates the M interfering lines using M orthogonalcodes and is less power-peaky in frequency than multi-line FDS. Under anaverage power constraint, multi-line FDS signaling strategy ispreferred. In multi-line FDS each line gets a separate frequency slotwithin each bin for transmission. The PSD s_(i) ^(o)(f) in each bin is Mtimes higher (or taller) than the corresponding PSD using EQPSDsignaling (see FIG. 18). Clearly, under a peak power constraint analternative orthogonal signaling scheme like multi-line CDS ispreferred. We can group together bins using multi-line FDS into onespectral region E_(MFDS). We can implement SS-CDS over this spectralregion E_(MFDS) such thatS _(i) ^(o)(f)⊥S _(i) ^(o)(f), ∀i≠j, oε{u,d}, and ∀fεE _(MFDS).  (111)

Note that implementation of SS-CDS cannot give perfectly orthogonalcodes; instead we have only codes with very low cross-correlation.However, use of CDS or multi-line CDS signaling yields similar capacity(in the limit as cross-correlation between codes→0) as FDS or multi-lineFDS schemes.

4.12.7 Multi-user Detector at Central Office

We have seen that self-interference is a major limiter in achievinghigher channel capacity. We can extend the work in previous Sections andconstruct a multi-user detector [21] at the central office that uses theself-interference for joint detection of each user (or line). In thissense the self-interference is not treated as only noise but can be usedas information to achieve further significant gains in capacity oftwisted pair lines.

Summary of Contributions

The key differences from the prior art are:

-   -   1. Increased capacity for xDSL lines using optimal and        suboptimal transmit spectra involving joint signaling schemes.    -   2. “Symmetrical” (or power complementary) upstream/downstream        optimal transmit spectrum for a xDSL line in presence of        self-NEXT, self-FEXT, AGN, and other interfering lines like T1,        HDSL, and ADSL using EQPSD and FDS signaling.    -   3. Fast near-optimal solution for the transmit spectrum which is        computationally very attractive and very easy to implement for        xDSL lines.    -   4. Spectral optimization gives good spectral compatibility with        other services (FDS better than CDS for spectral compatibility        under an average power constraint).    -   5. Dynamic transmit spectrum that adjusts automatically        according to the interference type.    -   6. Multi-line FDS signaling technique to combat self-FEXT.    -   7. Increased capacity for HDSL2, “GDSL”, and “VDSL2” lines using        multi-line FDS signaling when appropriate.    -   8. Increased capacity in generic xDSL lines when neighboring        lines have different channel, noise and interference        characteristics.    -   9. Concept of static estimation of interference values by        reading look-up table of the topology of the cables (which        self-interfering lines are where) at powerup. The        self-interference values can be estimated in this manner.        Dynamic measurement of interference values is done by        “listening” to the interference during powerup. (Subtract the        estimated self-interference from this measured interference to        get the different service interference.)    -   10. We can also interpret our results as capacity estimates        given a fixed margin in the presence of fixed interferers.        Final notes:    -   1. We have framed our work within the context of the HDSL2,        “GDSL”, and “VDSL2” transmission formats. However, our results        are more general, and apply to all channels that exhibit        crosstalk interference from neighboring channels. We summarize a        few channels where this technique could be potentially applied:        -   (a) Twisted pair lines (standard telephone lines)        -   (b) Untwisted pairs of copper lines        -   (c) Unpaired cables        -   (d) Coaxial cables        -   (e) Power lines        -   (f) Geophysical well-logging telemetry cables        -   (g) Wireless channels.    -   2. If a static mask is desired (e.g., for ease of        implementation), we propose that a thorough study be made of the        optimal solutions in different interference and noise scenarios        as proposed in this document and then a best static compromising        PSD mask be chosen.

REFERENCES

-   [1] S. McCaslin, “Performance and Spectral Compatibility of    MONET-PAM HDSL2 with Ideal Transmit Spectra-Preliminary Results,”    T1E1.4/97-307.-   [2] M. Rude, M. Sorbara, H. Takatori and G. Zimmerman, “A Proposal    for HDSL2 Transmission: OPTIS,” T1E1.4/97-238.-   [3] A. Sendonaris, V. Veeravalli and B. Aazhang, “Joint Signaling    Strategies for Approaching the Capacity of Twisted Pair Channels,”    IEEE Trans. Commun., vol. 46, no. 5, May 1998.-   [4] S. McCaslin and N. V. Bavel, “Performance and Spectral    Compatibility of MONET(R1) HDSL2 with Ideal Transmit    Spectra-Preliminary Results,” T1E1.4/97-412.-   [5] J. Girardeau, M. Rude, H. Takatori and G. Zimmerman, “Updated    OPTIS PSD Mask and Power Specification for HDSL2,” T1E1.4/97-435.-   [6] J. A. C. Bingham, “Multicarrier Modulation for Data    Transmission: An Idea Whose Time has Come,” IEEE Commun. Magazine,    May 1990.-   [7] G. Zimmerman, “Performance and Spectral Compatibility of OPTIS    HDSL2,” T1E1.4/97-237.-   [8] K. Kerpez, “Full-duplex 2B1Q Single-pair HDSL Performance and    Spectral Compatibility,” T1E1.4/95-127.-   [9] American National Standard for Telecommunications, “Network and    Customer Installation Interfaces—Asymmetric Digital Subscriber line    (ADSL) Metallic Interface,” T1.413-1995, Annex B.-   [10] American National Standard for Telecommunications, “Network and    Customer Installation Interfaces—Asymmetric Digital Subscriber Line    (ADSL) Metallic Interface,” T1.413-1995, Annex E.-   [11] G. Zimmerman, “Normative Text for Spectral Compatibility    Evaluations,” T1E1.4/97-180R1.-   [12] M. Barton and M. L. Honig, “Optimization of Discrete Multitone    to Maintain Spectrum Compatibility with Other Transmission Systems    on Twisted Copper Pairs,” IEEE J. Select. Areas Commun., vol. 13,    no. 9, pp. 1558-1563, December 1995.-   [13] K. J. Kerpez, “Near-End Crosstalk is almost Gaussian,” IEEE    Trans. Commun., vol. 41, no. 1, January 1993.-   (14] R. G. Gallager, “Information Theory and Reliable    Communication:” New York: Wiley, 1968.-   [15] I. Kalet, “The Multitone Channel,” IEEE Trans. Commun., vol.    37, no. 2, February 1989.-   [16] J. T. Aslanis and J. M. Cioffi, “Achievable Information Rates    on Digital Subscriber Loops: Limiting Information Rates with    Crosstalk Noise,” IEEE Trans. Commun., vol. 40, no. 2, February    1992.-   [17] P. S. Chow, J. M. Cioffi and J. A. C. Bingham, “A Practical    Discrete Multitone Transceiver Loading Algorithm for Data    Transmission over Spectrally Shaped Channels,” IEEE Trans. Commun.,    vol. 43, nos. 2/3/4, February/March/April 1995.-   [18] I. Kalet and S. Shamai (Shitz), “On the Capacity of a    Twisted-Wire Pair: Gaussian Model,” IEEE Trans. Commun., vol. 38,    no. 3, March 1990.-   [19] W. H. Press, S. A. Teukolsky, W. T. Vellerling and B. P.    Flannery, “Numerical recipes in C—The Art of Scientific Computing,”    Cambridge University Press, 2nd edition, 1997.-   [20] J. G. Proakis, “Digital Communications,” McGraw Hill, 3rd    edition, 1995-   [21] S. Verdu, “Recent-Progress in Multiuser Detection” in “Multiple    Access Communications,” Edited by N. Abramson IEEE press, 1993-   [22] R. Horst, P. M. Pardalos and N. V. Thoai, “Introduction to    Global Optimization,” Kluwer Academic Publishers, 1995

Glossary

-   ADSL: Asymmetrical digital subscriber line-   AGN: Additive Gaussian noise-   BER: Bit error rate (or probability)-   BT: Bridged tap-   CAP: Carrierless amplitude/pulse modulation-   CDMA: Code-division multiple access-   CDS: Code-division signaling-   CO: Central office-   CSA: Carrier serving area-   DFE: Decision feedback equalization-   DMT: Discrete multitone technology-   DSL: Digital subscriber line-   EQPSD: Equal power spectral density signaling-   FDS: Frequency division signaling-   FEXT: Far-end crosstalk-   “GDSL”: General digital subscriber line-   HDSL: High bit-rate digital subscriber line-   HDSL2: High bit-rate digital subscriber line 2-   ISDN: Integrated services digital network-   ISI: Intersymbol interference-   MFDS: Multi-line Frequency division signaling-   NEXT: Near-end crosstalk-   PAM: Pulse amplitude modulation-   POTS: Plain old telephone services-   PSD: Power spectral density-   QAM: Quadrature amplitude modulation-   SNR: Signal to noise ratio-   SS-CDS: Spread spectrum code-division signaling-   T1: Transmission 1 standard-   TDS: Time division signaling-   VDSL: Very high bit-rate DSL-   “VDSL2”: Very high bit-rate DSL 2-   xDSL: Any generic DSL service

Notation

-   ⊥: Orthogonal-   ∪: Union-   A: Kind of service, such as ADSL, HDSL, HDSL2, VDSL, etc.-   B: Channel transmission bandwidth-   C: Channel capacity or line capacity-   D: Difference between two capacities-   E: Spectral region-   F: Magnitude squared Far-end crosstalk (self-FEXT) transfer function    in a single bin-   G: Signal to noise ratio (SNR) in a single bin-   H: Magnitude squared channel transfer function in a single bin-   J: Kind of signaling scheme, such as EQPSD, FDS, multi-line FDS,    etc.-   K: Total number of bins within channel transmission bandwidth-   L: Function of line parameters (G, F, X, H) in a single bin; it is    always a positive quantity-   M: Number of interfering lines carrying the same service-   N: Total additive Gaussian noise (AGN) power plus total different    service interference-   P: Power-   Q: Constraining PSD mask-   R: Receiver-   S: Power spectral density (PSD)-   T: Transmitter-   U: Positive quantity equal to Y+Z+N-   V: Positive quantity equal to Y+Z+N+S-   W: Bandwidth of a bin or a subchannel-   X: Magnitude squared Near-end crosstalk (self-NEXT) transfer    function in a single bin-   Y: Part of crosstalk power that couples into another service line-   Z: Part of crosstalk power that couples into another service line-   a: An amplitude level of a transmit spectrum-   b: An amplitude level of a transmit spectrum-   c: Capacity of a bin or a subchannel-   d: Downstream direction-   f: Frequency-   i: Line number-   j: Line number-   k: Bin index-   n: Fraction to choose power distribution, 0≦n≦1-   o: Direction index oε{u, d}-   u: Upstream direction-   C_(i) ^(o): Capacity of line i in transmission direction o-   C^(o): Capacity of a line in direction o-   C_(i): Capacity of a line i-   E_(FDS): Spectral region employing FDS signaling-   E_(MFDS): Spectral region employing multi-line FDS signaling-   F_(i,k): Magnitude squared self-FEXT transfer function on line i and    bin k-   F_(i): Magnitude squared self-ET transfer function of line i in a    single bin-   G_(i): Ratio of signal power in line i to noise power in line 1 in a    single bin-   H_(i,k): Magnitude squared channel transfer function of line i and    bin k-   H_(i): Magnitude squared channel transfer function of line i in a    single bin-   N_(o)(f): Channel noise-   N_(i): AGN plus different service interference on line i-   P_(mi): Power in positive frequency range ([0, W]) of a single bin    of line i-   P_(m): Power in positive frequency range ([0, W]) of a single bin-   P_(max): Total average power over the entire frequency range ([−B,    B]) of the channel-   Q^(o)(f): Constraining PSD mask in direction o-   R_(A): Achievable rate in a single bin or subchannel-   R_(i) ^(o): Receiver on line i in direction o-   S_(i) ^(o)(f): PSD of line i in direction o-   S^(o)(f): PSD of a line in direction o-   T_(i) ^(o): Transmitter on line i in direction o-   X_(i,k): Magnitude squared self-NEXT transfer function on line i and    bin k-   X_(i): Magnitude squared self-NEXT transfer function on line i in a    single bin-   c_(i,J) ^(o): Capacity of a single bin of line i using signaling    scheme J.-   c_(i) ^(o): Capacity of a single bin of line i in direction o-   c^(o): Capacity of a single bin in direction o-   s_(i) ^(o)(f): PSD in a single bin of line i in direction o-   s^(o)(f): PSD in a single bin in direction o

1. A method for communicating data on a set of communications channelslimited by predetermined peak power constraints in frequency, andwherein each communications channel in the set of communicationschannels is subject to interference, the method comprising: determiningchannel transfer functions of the set of communications channels;determining interference characteristics of the set of communicationschannels; determining transmit spectra for the set of communicationschannels in response to the channel transfer functions, the interferencecharacteristics, and the predetermined peak power constraints infrequency, wherein said determining the transmit spectra comprises usinga peak-constrained water-filling technique to determine power spectraldensity functions for the set of communications channels; andtransmitting data on the set of communications channels with spectralpower distributions given by the transmit spectra.
 2. The method ofclaim 1, wherein one or more of the communications channels are furtherlimited to a predetermined average power, and wherein one or more of thetransmit spectra are determined in response to the predetermined averagepower.
 3. The method of claim 1, wherein said transmitting data on theset of communications channels comprises transmitting data using onetype of service on all communications channels in the set ofcommunications channels.
 4. The method claim 1, wherein thecommunications channels in the set of communications channels carry onetype of service from the group consisting of: xDSL, ISDN, T1, T2, Tn,cable modem, and wireless modem.
 5. The method claim 1, wherein the setof communications channels is a subset of a set of communicationschannels carrying one type of service.
 6. The method of claim 1, whereinat least one communications channel in the set of communicationschannels is a twisted-pair line.
 7. The method of claim 1, wherein saidtransmitting data on the set of communications channels comprisestransmitting the data in discrete multi-tone signals.
 8. The method ofclaim 1, wherein the predetermined peak power constraint in frequency isa constraining static PSD mask.
 9. The method of claim 1, wherein saiddetermining the transmit spectra comprises determining one upstreamtransmit spectrum for the set of communications channels and onedownstream transmit spectrum for the set of communications channels. 10.The method of claim 1, wherein said determining the transmit spectracomprises determining one transmit spectrum for each one of thecommunications channels in the set of communications channels.
 11. Themethod of claim 1, wherein said determining the transmit spectracomprises determining one upstream transmit spectrum for each one of thecommunications channels in the set of communications channels anddetermining one downstream transmit spectrum for each one of thecommunications channels in the set of communications channels.
 12. Themethod of claim 1, wherein said determining interference characteristicscomprises: determining an amount of uncorrelated interference into theset of communications channels; and determining an amount of cross-talkcoupling among communications channels in the set of communicationschannels; wherein the transmit spectra are determined in response to theuncorrelated interference and the amount of cross-talk coupling.
 13. Themethod of claim 1, wherein said determining the transmit spectracomprises determining one or more upstream transmit spectra in responseto an upstream portion of the predetermined peak power constraints infrequency and determining one or more downstream transmit spectra inresponse to a downstream portion of the predetermined peak powerconstraints in frequency.
 14. The method of claim 1, wherein saiddetermining the transmit spectra comprises determining one or moreupstream transmit spectra in response to a predetermined averageupstream power and determining one or more downstream transmit spectrain response to a predetermined average downstream power.
 15. The methodof claim 1, wherein the interference includes additive Gaussian noise(AGN); wherein the transmit spectra are determined in response to theamount of AGN.
 16. The method of claim 1, wherein the interferenceincludes different service interference (DSIN); wherein the transmitspectra are determined in response to the amount of DSIN.
 17. The methodof claim 1, wherein said determining the channel transfer functionscomprises receiving the channel transfer functions.
 18. The method ofclaim 1, wherein said determining the channel transfer functionscomprises measuring the channel transfer functions.
 19. The method ofclaim 1, wherein said determining the channel transfer functions isperformed in response to a power-up, or at regular intervals in time, orin response to temperature changes.
 20. The method of claim 1, whereinsaid determining interference characteristics and said determining thetransmit spectra are dynamically performed each time a data transfer isinitiated.
 21. The method of claim 1, wherein the transmit spectra aredetermined so that the communications channels have equal upstream anddownstream capacities.
 22. The method of claim 1, wherein the transmitspectra are determined so that the communications channels have equalupstream and downstream performance margins.
 23. The method of claim 1,wherein the transmit spectra for the set of communications channels arespectrally compatible.
 24. The method of claim 1, wherein the transmitspectra for the set of communications channels are spectrally compatibleand are configured to substantially maximize a data transmission ratefor the communications channels.
 25. A method for communicating data ona set of communications channels limited by predetermined peak powerconstraints in frequency, and wherein each communications channel in theset of communications channels is subject to interference, the methodcomprising: determining channel transfer functions of the set ofcommunications channels; determining interference characteristics of theset of communications channels; determining transmit spectra for the setof communications channels in response to the channel transferfunctions, the interference characteristics, and the predetermined peakpower constraints in frequency; and transmitting data on the set ofcommunications channels with spectral power distributions given by thetransmit spectra, wherein said determining the transmit spectracomprises determining one transmit spectrum for each one communicationschannel in the set of communications channels, wherein said determiningone transmit spectrum for one communications channel comprises: (a)determining a preliminary transmit spectrum S(f) in response to theinterference characteristics and the channel transfer function for theone communications channel; (b) comparing the preliminary transmitspectrum S(f) to the predetermined peak power constraint Q(f); and (c)in spectral regions where S(f)>Q(f), modifying the preliminary transmitspectrum S(f) so that S(f)=Q(f) to generate the transmit spectrum forthe one communications channel.
 26. The method of claim 25, wherein saiddetermining the preliminary transmit spectrum S(f) comprises determininga substantially optimized transmit spectrum.
 27. A method forcommunicating data on a set of communications channels limited bypredetermined peak power constraints in frequency, and wherein eachcommunications channel in the set of communications channels is subjectto interference, the method comprising: determining channel transferfunctions of the set of communications channels; determininginterference characteristics of the set of communications channels;determining transmit spectra for the set of communications channels inresponse to the channel transfer functions, the interferencecharacteristics, and the predetermined peak power constraints infrequency; and transmitting data on the set of communications channelswith spectral power distributions given by the transmit spectra, whereinsaid determining the transmit spectra comprises determining two transmitspectra for each one communications channel in the set of communicationschannels, wherein for each communications channel the two transmitspectra are used for signaling in opposite directions along thecommunications channel, and wherein said determining two transmitspectra for one communications channel comprises: determining asubstantially optimized transmit spectrum S^(u)(f) for signaling in afirst direction in response to the channel transfer function of the onecommunications channel and in response to the interferencecharacteristics; determining a first transmit spectrum S_(opt) ^(u)(f)in the two transmit spectra from the substantially optimized transmitspectrum S^(u)(f) and from the predetermined peak power constraint infrequency Q(f), wherein S_(opt) ^(u)(f)=S^(u)(f) in spectral regionswhere S^(u)(f)<=Q(f) and wherein S_(opt) ^(u)(f)=Q(f) in spectralregions where S^(u)(f)>Q(f), and wherein the first transmit spectrumS_(opt) ^(u)(f) is useable for transmitting data on the onecommunications channel in the first direction; and determining a secondtransmit spectrum S_(opt) ^(d)(f) in the two transmit spectra from thefirst transmit spectrum S_(opt) ^(u)(f), wherein the second transmitspectrum S_(opt) ^(u)(f) is complementary to the first transmit spectrumS_(opt) ^(u)(f), and wherein the second transmit spectrum S_(opt)^(d)(f) is useable for transmitting data on the one communicationschannel in a second direction.
 28. The method of claim 27, wherein saiddetermining the substantially optimized transmit spectrum comprisesusing a water-filling technique.
 29. The method of claim 27, whereinsaid determining interference characteristics includes determining anamount of near-end cross-talk interference among channels in the set ofcommunications channels; and wherein said determining the substantiallyoptimized transmit spectrum S^(u)(f) comprises determining a transmitspectrum with at least one spectral region of FDS signaling in responseto the near-end cross-talk interference.
 30. The method of claim 27,wherein said determining interference characteristics includesdetermining an amount of near-end cross-talk interference among channelsin the set of communications channels; and wherein said determining thefirst transmit spectrum and said determining the second transmitspectrum comprise determining the first and second transmit spectra suchthat the first and second transmit spectra are orthogonally separated inat least one spectral region in response to the near-end cross-talkinterference.
 31. The method of claim 27, wherein said determininginterference characteristics includes determining an amount of far-endcross-talk interference among channels in the set of communicationschannels; and wherein said determining the substantially optimizedtransmit spectrum S^(u)(f) comprises determining a transmit spectrumwith at least one spectral region of multi-line FDS signaling inresponse to the far-end cross-talk interference.
 32. The method of claim27, wherein said determining interference characteristics includesdetermining an amount of far-end cross-talk interference among channelsin the set of communications channels; and wherein said determining thetransmit spectra comprises determining the transmit spectra such thatthe communications channels in the set of communications channels areorthogonally separated in at least one spectral region in response tothe far-end cross-talk interference.
 33. A method for communicating dataon a set of communications channels limited by predetermined peak powerconstraints in frequency, and wherein each communications channel in theset of communications channels is subject to interference, the methodcomprising: determining channel transfer functions of the set ofcommunications channels; determining interference characteristics of theset of communications channels; determining transmit spectra for the setof communications channels in response to the channel transferfunctions, the interference characteristics, and the predetermined peakpower constraints in frequency; and transmitting data on the set ofcommunications channels with spectral power distributions given by thetransmit spectra, wherein the predetermined peak power constraint infrequency includes a first peak power constraint in frequency Q^(u)(f)for signaling in a first direction and a second peak power constraint infrequency Q^(d)(f) for signaling in a second direction, and wherein thecommunications channels are further limited by a predetermined totalpower constraint, wherein said determining the transmit spectracomprises determining two transmit spectra for each one communicationschannel in the set of communications channels, wherein for eachcommunications channel the two transmit spectra are used for signalingin opposite directions along the communications channel, and whereinsaid determining the two transmit spectra for one communications channelcomprises: determining a substantially optimized transmit spectrumS^(u)(f) for signaling in a first direction in response to a channeltransfer function of the one communications channel and in response tothe interference characteristics; determining a first constrainedtransmit spectrum S_(opt) ^(u)(f) in response to the substantiallyoptimized transmit spectrum S^(u)(f) and in response to a combined peakpower constraint Q(f)=max(Q^(u)(f), Q^(d)(f)), wherein S_(opt)^(u)(f)=S^(u)(f) in spectral regions where S^(u)(f)<=Q(f) and whereinS_(opt) ^(u)(f)=Q(f) in spectral regions where S^(u)(f)>Q(f);determining a second constrained transmit spectrum S_(opt) ^(d)(f) inresponse to the first constrained transmit spectrum S_(opt) ^(u)(f),wherein the second constrained transmit spectrum S_(opt) ^(d)(f) iscomplementary to the first constrained transmit spectrum S_(opt)^(u)(f); merging the first and second constrained transmit spectra toform a combined constrained transmit spectrum S_(opt)(f); determining afirst transmit spectrum S₁ ^(u)(f) in the two transmit spectra from thecombined constrained transmit spectrum S_(opt)(f) in response toQ^(u)(f) and Q^(d)(f); determining a second transmit spectrum S₁ ^(d)(f)in the two transmit spectra from the combined constrained transmitspectrum S_(opt)(f) in response to Q(f) and Q^(d)(f); if a total powerof the first transmit spectrum violates the predetermined total powerconstraint, modifying the first transmit spectrum S₁ ^(u)(f) to reducethe total power of the first transmit spectrum; and if a total power ofthe second transmit spectrum violates the predetermined total powerconstraint, modifying the second transmit spectrum S₁ ^(d)(f) to reducethe total power of the second transmit spectrum.
 34. The method of claim33, wherein said determining the substantially optimized transmitspectrum comprises using a water-filling technique to calculate thesubstantially optimized transmit spectrum.
 35. The method of claim 33,wherein said determining interference characteristics includesdetermining an amount of near-end cross-talk interference among channelsin the set of communications channels; and wherein said determining thesubstantially optimized transmit spectrum S^(u)(f) comprises determininga transmit spectrum with at least one spectral region of FDS signalingin response to the near-end cross-talk interference.
 36. The method ofclaim 33, wherein said determining interference characteristics includesdetermining an amount of near-end cross-talk interference among channelsin the set of communications channels; and wherein said determining thefirst constrained transmit spectrum and said determining the secondconstrained transmit spectrum comprise determining the first and secondconstrained transmit spectra such that the first and second constrainedtransmit spectra are orthogonally separated in at least one spectralregion in response to the near-end cross-talk interference.
 37. Themethod of claim 33, wherein said determining interferencecharacteristics includes determining an amount of far-end cross-talkinterference among channels in the set of communications channels; andwherein said determining the substantially optimized transmit spectrumS^(u)(f) comprises determining a transmit spectrum with at least onespectral region of multi-line FDS signaling.
 38. The method of claim 33,wherein said determining interference characteristics includesdetermining an amount of far-end cross-talk interference among channelsin the set of communications channels; and wherein said determining thetransmit spectra comprises determining the transmit spectra such thateach communications channel in the set of communications channels isorthogonally separated from other communications channels in the set ofcommunications channels in at least one spectral region in response tothe far-end cross-talk interference.
 39. The method of claim 33, whereinthe second constrained transmit spectrum is equal to the firstconstrained transmit spectrum in spectral regions of EQPSD signaling inthe first constrained transmit spectrum.
 40. The method of claim 33,wherein the second constrained transmit spectrum is orthogonallyseparated from the first constrained transmit spectrum in spectralregions of FDS signaling in the first constrained transmit spectrum. 41.The method of claim 33, wherein the second constrained transmit spectrumis equal to the first constrained transmit spectrum in spectral regionsof multi-line FDS signaling in the first constrained transmit spectrum,and wherein each communications channel in the set of communicationschannels is orthogonally separated from other communications channels inthe set of communications channels in spectral regions of multi-line FDSsignaling in the first constrained transmit spectrum.
 42. The method ofclaim 33, wherein said merging the first and second constrained transmitspectra comprises constructing the combined constrained transmitspectrum S_(opt)(f) such that, in spectral regions of EQPSD signaling inthe first constrained transmit spectrum, the combined constrainedtransmit spectrum S_(opt)(f) is equal to the first constrained transmitspectrum.
 43. The method of claim 33, wherein said merging the first andsecond constrained transmit spectra comprises constructing the combinedconstrained transmit spectrum S_(opt)(f) such that, in spectral regionsof FDS signaling in the first constrained transmit spectrum, thecombined constrained transmit spectrum S_(opt)(f) is equal to themaximum of the first and second constrained transmit spectra.
 44. Themethod of claim 33, wherein said merging the first and secondconstrained transmit spectra comprises constructing the combinedconstrained transmit spectrum S_(opt)(f) such that, in spectral regionsof multi-line FDS signaling in the first constrained transmit spectrum,the combined constrained transmit spectrum S_(opt)(f) is equal to themaximum of constrained transmit spectra for the communications channelsin the set of communications channels.
 45. The method of claim 33,wherein said modifying the first transmit spectrum S₁ ^(u)(f) to reducethe total power of the first transmit spectrum comprises modifying thefirst transmit spectrum S₁ ^(u)(f) and the second transmit spectrum S₁^(d)(f) so that the total power of the first transmit spectrum isdecreased by an amount of power and so that the total power of thesecond transmit spectrum is increased by the amount of power.
 46. Themethod of claim 33, wherein said modifying the second transmit spectrumS₁ ^(d)(f) to reduce the total power of the second transmit spectrumcomprises modifying the second transmit spectrum S₁ ^(d)(f) and thefirst transmit spectrum S₁ ^(u)(f) so that the total power of the secondtransmit spectrum is decreased by an amount of power and so that thetotal power of the first transmit spectrum is increased by the amount ofpower.
 47. A method for communicating data on a set of communicationschannels limited by predetermined peak power constraints in frequency,and wherein each communications channel in the set of communicationschannels is subject to interference, the method comprising: determiningchannel transfer functions of the set of communications channels;determining interference characteristics of the set of communicationschannels; determining transmit spectra for the set of communicationschannels in response to the channel transfer functions, the interferencecharacteristics, and the predetermined peak power constraints infrequency; and transmitting data on the set of communications channelswith spectral power distributions given by the transmit spectra, whereinsaid determining the interference characteristics comprises determininga noise power spectral density (PSD) for each one of the communicationschannels in the set of communications channels; wherein said determiningthe transmit spectra comprises: determining a first transmit spectrumS₁(f) for each one of the communications channels in the set ofcommunications channels, wherein the first transmit spectrum for onecommunications channel has the form given by the following equation;${{S_{1}(f)} = {\lambda - \frac{I(f)}{{{H_{C}(f)}}^{2}}}},$ whereinI(f) is the noise PSD for the one communications channel, whereinH_(C)(f) is a channel transfer function for the one communicationschannel, and wherein λ is determined in response to a predeterminedaverage power for the one communications channel.
 48. A method forcommunicating data on a communications channel, wherein thecommunications channel is limited by a predetermined peak powerconstraint in frequency, and wherein the communications channel issubject to interference, the method comprising: determining a channeltransfer function of the communications channel; determininginterference characteristics of the communications channel; determininga transmit spectrum in response to the channel transfer function, theinterference characteristics, and the predetermined peak powerconstraint in frequency, wherein said determining the transmit spectrumcomprises using a peak-constrained water-filling technique to determinea power spectral density function; and transmitting data on thecommunications channel with a spectral power distribution given by thetransmit spectrum.
 49. The method of claim 48, wherein thecommunications channel is further limited to a predetermined averagepower, and wherein the transmit spectrum is determined in response tothe predetermined average power.
 50. The method of claim 48, wherein thepredetermined peak power constraint in frequency is a constrainingstatic PSD mask.
 51. The method of claim 48, wherein said determiningthe transmit spectrum comprises determining an upstream transmitspectrum in response to an upstream portion of the predetermined peakpower constraint in frequency and determining a downstream transmitspectrum in response to a downstream portion of the predetermined peakpower constraint in frequency.
 52. The method of claim 48, wherein saiddetermining the transmit spectrum comprises determining an upstreamtransmit spectrum in response to a predetermined average upstream powerand determining a downstream transmit spectrum in response to apredetermined average downstream power.
 53. The method of claim 48,wherein the interference includes additive Gaussian noise (AGN); whereinthe transmit spectrum is determined in response to the amount of AGN.54. The method of claim 48, wherein the interference includes differentservice interference (DSIN); wherein the transmit spectrum is determinedin response to the amount of DSIN.
 55. The method of claim 48, whereinsaid determining the channel transfer function comprises receiving thechannel transfer function.
 56. The method of claim 48, wherein saiddetermining the channel transfer function comprises measuring thechannel transfer function.
 57. The method of claim 48, wherein saiddetermining the channel transfer function is performed in response to apower-up, or at regular intervals in time, or in response to temperaturechanges.
 58. The method of claim 48, wherein said determininginterference characteristics and said determining the transmit spectrumare dynamically performed each time a data transfer is initiated. 59.The method of claim 48, wherein the transmit spectrum is determined sothat the communications channel has equal upstream and downstreamcapacities.
 60. The method of claim 48, wherein the transmit spectrum isdetermined so that the communications channel has equal upstream anddownstream performance margins.
 61. The method of claim 48, wherein thetransmit spectrum is spectrally compatible with the one or more othercommunications channels.
 62. The method of claim 48, wherein thetransmit spectrum is configured to substantially maximize a datatransmission rate for the communications channel and is spectrallycompatible with the one or more other communications channels.
 63. Themethod of claim 48, wherein said determining interferencecharacteristics and said transferring the data are performed a pluralityof times; wherein said determining interference characteristics and saiddetermining the transmit spectrum are performed each time a datatransfer is initiated.
 64. The method of claim 48, further comprising:repeating said determining interference characteristics and saiddetermining the transmit spectrum during the data transfer to produce anew transmit spectrum; wherein the new transmit spectrum is used duringa remainder of the data transfer.
 65. The method of claim 48, whereinsaid transmitting data on the communications channel comprisestransmitting data on a twisted-pair line.
 66. The method of claim 48,wherein said transmitting data on the communications channel comprisestransmitting the data on the communications channel in a discretemulti-tone signal.
 67. A method for communicating data on acommunications channel, wherein the communications channel is limited bya predetermined peak power constraint in frequency, and wherein thecommunications channel is subject to interference, the methodcomprising: determining a channel transfer function of thecommunications channel; determining interference characteristics of thecommunications channel; determining a transmit spectrum in response tothe channel transfer function, the interference characteristics, and thepredetermined peak power constraint in frequency; and transmitting dataon the communications channel with a spectral power distribution givenby the transmit spectrum, wherein said determining the transmit spectrumcomprises: (a) determining a preliminary transmit spectrum S(f) inresponse to the channel transfer function and the interferencecharacteristics; (b) comparing the preliminary transmit spectrum S(f) tothe predetermined peak power constraint Q(f); and (c) in spectralregions where S(f)>Q(f), modifying the preliminary transmit spectrumS(f) so that S(f)=Q(f).
 68. The method of claim 67, wherein saiddetermining preliminary transmit spectrum S(f) comprises determining asubstantially optimized transmit spectrum.
 69. A method forcommunicating data on a communications channel, wherein thecommunications channel is limited by a predetermined peak powerconstraint in frequency, and wherein the communications channel issubject to interference, the method comprising: determining a channeltransfer function of the communications channel; determininginterference characteristics of the communications channel; determininga transmit spectrum in response to the channel transfer function, theinterference characteristics, and the predetermined peak powerconstraint in frequency; and transmitting data on the communicationschannel with a spectral power distribution given by the transmitspectrum, wherein the communications channel is further limited to apredetermined average power, and wherein said determining the transmitspectrum comprises: (a) determining a first transmit spectrum S₁(f) inresponse to the channel transfer function and the interferencecharacteristics; (b) comparing the first transmit spectrum S₁(f) to thepredetermined peak power constraint Q(f); (c) determining a secondtransmit spectrum S₂(f), wherein S₂(f)=S₁(f) in spectral regions whereS₁(f)<=Q(f) and wherein S₂(f)=Q(f) in spectral regions where S₁(f)>Q(f);(d) determining an average power of the second transmit spectrum S₂(f);and (e) if the average power of the second transmit spectrum S₂(f) isnot substantially equal to the predetermined average power, modifyingthe first transmit spectrum S₁(f) and repeating said steps (b)-(e). 70.The method of claim 69, wherein said determining the first transmitspectrum S₁(f) comprises determining a substantially optimized transmitspectrum.
 71. The method of claim 69, wherein said determining the firsttransmit spectrum S₁(f) comprises determining a substantially optimizedtransmit spectrum; wherein said determining the first transmit spectrumS₁(f) is performed in response to a set of one or more adjustableparameters, and wherein said modifying the first transmit spectrum S₁(f)comprises changing one or more of the adjustable parameters andredetermining the first transmit spectrum S₁(f) in response to theadjustable parameters.
 72. The method of claim 69, wherein saiddetermining the first transmit spectrum S₁(f) comprises determining asubstantially optimized transmit spectrum; and wherein said modifyingthe first transmit spectrum S₁(f) comprises adding or subtracting anoffset to the first transmit spectrum S₁(f).
 73. The method of claim 69,wherein said determining the interference characteristics of thecommunications channel comprises determining a noise power spectraldensity (PSD) I(f) for the communications channel; wherein saiddetermining the first transmit spectrum S₁(f) comprises determining atransmit spectrum with the form given by the following equation;${{S_{1}(f)} = {\lambda - \frac{I(f)}{{{H_{C}(f)}}^{2}}}},$ whereinH_(C)(f) is the channel transfer function and wherein λ is a constant;wherein said modifying the first transmit spectrum S₁(f) compriseschanging the constant λ to minimize a difference between the averagepower of the second transmit spectrum S₂(f) and the predeterminedaverage power.
 74. The method of claim 69, wherein said determining theinterference characteristics of the communications channel comprisesdetermining an amount of channel noise N_(o)(f) into the communicationschannel, an amount of near-end cross-talk interference from acommunications channel carrying a different type of service DS_(N)(f)into the communications channel, and an amount of far-end cross-talkinterference from a communications channel carrying a different type ofservice DS_(F)(f) into the communications channel; wherein saiddetermining the first transmit spectrum S₁(f) comprises determining atransmit spectrum with the form given by the following equation;${{S_{1}(f)} = {\lambda - \frac{{N_{o}(f)} + {{DS}_{N}(f)} + {{DS}_{F}(f)}}{{{H_{C}(f)}}^{2}}}},$wherein H_(C)(f) is the channel transfer function and wherein λ is aconstant; wherein said modifying the first transmit spectrum S₁(f)comprises changing the constant λ to minimize a difference between theaverage power of the second transmit spectrum S₂(f) and thepredetermined average power.
 75. A method for determining a transmitspectrum for a communications channel, wherein the transmit spectrum islimited by a predetermined peak power constraint in frequency, andwherein the communications channel is subject to interference, themethod comprising: determining a channel transfer function of thecommunications channel; determining interference characteristics of thecommunications channel; determining the transmit spectrum in response tothe channel transfer function, the interference characteristics, and thepredetermined peak power constraint in frequency, wherein saiddetermining the transmit spectrum comprises using a peak-constrainedwater-filling technique to determine a power spectral density function;wherein the transmit spectrum is useable in communicating data on thecommunications channel.
 76. The method of claim 75, further comprising:transmitting data on the communications channel using the transmitspectrum.
 77. The method of claim 76, wherein said transmitting data onthe communications channel comprises transmitting data on a twisted-pairline.
 78. The method of claim 76, wherein said transmitting data on thecommunications channel comprises transmitting the data on thecommunications channel in a discrete multi-tone signal.
 79. The methodof claim 75, wherein the transmit spectrum is further limited to apredetermined average power on the communications channel, and whereinthe transmit spectrum is determined in response to the predeterminedaverage power on the communications channel.
 80. A method fordetermining transmit spectra for use in communicating data on a set ofcommunications channels, wherein each communications channel in the setof communications channels is limited by a predetermined peak powerconstraint in frequency, and wherein each communications channel in theset of communications channels is subject to interference, the methodcomprising: determining channel transfer functions of the set ofcommunications channels; determining interference characteristics of theset of communications channels; and determining the transmit spectra forthe set of communications channels in response to the channel transferfunctions, the interference characteristics, and the predetermined peakpower constraint in frequency, wherein said determining the transmitspectra comprises using a peak-constrained water-filling technique todetermine power spectral density functions for the set of communicationschannels; wherein the transmit spectra are useable in communicating dataon the communications channels.
 81. The method of claim 80, furthercomprising: transmitting data on the communications channels using thetransmit spectra.
 82. The method of claim 81, wherein said transmittingdata on the communications channels comprises transmitting the data onthe communications channels in a discrete multi-tone signal.
 83. Themethod of claim 80, wherein each communications channel in the set ofcommunications channels is further limited to a predetermined averagepower, and wherein the transmit spectra are determined in response tothe predetermined average power.
 84. The method of claim 80, wherein atleast one communications channel in the set of communications channelsis a twisted-pair line.